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Properties

Label 4-10e2-1.1-c27e2-0-2
Degree 44
Conductor 100100
Sign 11
Analytic cond. 2133.102133.10
Root an. cond. 6.795996.79599
Motivic weight 2727
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.63e4·2-s − 4.70e6·3-s + 2.01e8·4-s − 2.44e9·5-s − 7.70e10·6-s − 5.71e10·7-s + 2.19e12·8-s + 1.11e13·9-s − 4.00e13·10-s + 1.69e14·11-s − 9.46e14·12-s − 1.03e15·13-s − 9.36e14·14-s + 1.14e16·15-s + 2.25e16·16-s − 5.04e16·17-s + 1.83e17·18-s + 4.46e17·19-s − 4.91e17·20-s + 2.68e17·21-s + 2.78e18·22-s − 6.64e18·23-s − 1.03e19·24-s + 4.47e18·25-s − 1.70e19·26-s − 3.70e19·27-s − 1.15e19·28-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.70·3-s + 3/2·4-s − 0.894·5-s − 2.40·6-s − 0.223·7-s + 1.41·8-s + 1.46·9-s − 1.26·10-s + 1.48·11-s − 2.55·12-s − 0.951·13-s − 0.315·14-s + 1.52·15-s + 5/4·16-s − 1.23·17-s + 2.07·18-s + 2.43·19-s − 1.34·20-s + 0.379·21-s + 2.09·22-s − 2.74·23-s − 2.40·24-s + 3/5·25-s − 1.34·26-s − 1.75·27-s − 0.334·28-s + ⋯

Functional equation

Λ(s)=(100s/2 ΓC(s)2 L(s)=( Λ(28−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(28-s) \end{aligned}
Λ(s)=(100s/2 ΓC(s+27/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\C}(s+27/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 100100    =    22⋅522^{2} \cdot 5^{2}
Sign: 11
Analytic conductor: 2133.102133.10
Root analytic conductor: 6.795996.79599
Motivic weight: 2727
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 100, ( :27/2,27/2), 1)(4,\ 100,\ (\ :27/2, 27/2),\ 1)

Particular Values

L(14)L(14) == 00
L(12)L(\frac12) == 00
L(292)L(\frac{29}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 (1−p13T)2 ( 1 - p^{13} T )^{2}
5C1C_1 (1+p13T)2 ( 1 + p^{13} T )^{2}
good3D4D_{4} 1+1567652pT+44989959806p5T2+1567652p28T3+p54T4 1 + 1567652 p T + 44989959806 p^{5} T^{2} + 1567652 p^{28} T^{3} + p^{54} T^{4}
7D4D_{4} 1+8169291644pT+53908827334167324402p4T2+8169291644p28T3+p54T4 1 + 8169291644 p T + 53908827334167324402 p^{4} T^{2} + 8169291644 p^{28} T^{3} + p^{54} T^{4}
11D4D_{4} 1−1403730832224p2T+ 1 - 1403730832224 p^{2} T + 23 ⁣⋯ ⁣2623\!\cdots\!26p2T2−1403730832224p29T3+p54T4 p^{2} T^{2} - 1403730832224 p^{29} T^{3} + p^{54} T^{4}
13D4D_{4} 1+79930383943292pT+ 1 + 79930383943292 p T + 87 ⁣⋯ ⁣0287\!\cdots\!02p2T2+79930383943292p28T3+p54T4 p^{2} T^{2} + 79930383943292 p^{28} T^{3} + p^{54} T^{4}
17D4D_{4} 1+50445145609767948T+ 1 + 50445145609767948 T + 38 ⁣⋯ ⁣2238\!\cdots\!22T2+50445145609767948p27T3+p54T4 T^{2} + 50445145609767948 p^{27} T^{3} + p^{54} T^{4}
19D4D_{4} 1−446863730094123400T+ 1 - 446863730094123400 T + 57 ⁣⋯ ⁣6257\!\cdots\!62pT2−446863730094123400p27T3+p54T4 p T^{2} - 446863730094123400 p^{27} T^{3} + p^{54} T^{4}
23D4D_{4} 1+289013388361901172pT+ 1 + 289013388361901172 p T + 39 ⁣⋯ ⁣8239\!\cdots\!82p2T2+289013388361901172p28T3+p54T4 p^{2} T^{2} + 289013388361901172 p^{28} T^{3} + p^{54} T^{4}
29D4D_{4} 1+47997399844470169380T+ 1 + 47997399844470169380 T + 25 ⁣⋯ ⁣1825\!\cdots\!18T2+47997399844470169380p27T3+p54T4 T^{2} + 47997399844470169380 p^{27} T^{3} + p^{54} T^{4}
31D4D_{4} 1−33574730315678836744T− 1 - 33574730315678836744 T - 51 ⁣⋯ ⁣9451\!\cdots\!94T2−33574730315678836744p27T3+p54T4 T^{2} - 33574730315678836744 p^{27} T^{3} + p^{54} T^{4}
37D4D_{4} 1+ 1 + 13 ⁣⋯ ⁣0813\!\cdots\!08T+ T + 39 ⁣⋯ ⁣8239\!\cdots\!82T2+ T^{2} + 13 ⁣⋯ ⁣0813\!\cdots\!08p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
41D4D_{4} 1+ 1 + 11 ⁣⋯ ⁣5611\!\cdots\!56T− T - 12 ⁣⋯ ⁣5412\!\cdots\!54T2+ T^{2} + 11 ⁣⋯ ⁣5611\!\cdots\!56p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
43D4D_{4} 1+ 1 + 39 ⁣⋯ ⁣3639\!\cdots\!36T+ T + 13 ⁣⋯ ⁣3813\!\cdots\!38T2+ T^{2} + 39 ⁣⋯ ⁣3639\!\cdots\!36p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
47D4D_{4} 1+ 1 + 32 ⁣⋯ ⁣2832\!\cdots\!28T+ T + 22 ⁣⋯ ⁣2222\!\cdots\!22T2+ T^{2} + 32 ⁣⋯ ⁣2832\!\cdots\!28p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
53D4D_{4} 1+ 1 + 33 ⁣⋯ ⁣9633\!\cdots\!96T+ T + 74 ⁣⋯ ⁣7874\!\cdots\!78T2+ T^{2} + 33 ⁣⋯ ⁣9633\!\cdots\!96p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
59D4D_{4} 1+ 1 + 34 ⁣⋯ ⁣6034\!\cdots\!60T+ T + 11 ⁣⋯ ⁣3811\!\cdots\!38T2+ T^{2} + 34 ⁣⋯ ⁣6034\!\cdots\!60p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
61D4D_{4} 1− 1 - 22 ⁣⋯ ⁣8422\!\cdots\!84T+ T + 21 ⁣⋯ ⁣0621\!\cdots\!06T2− T^{2} - 22 ⁣⋯ ⁣8422\!\cdots\!84p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
67D4D_{4} 1+ 1 + 39 ⁣⋯ ⁣2839\!\cdots\!28T+ T + 44 ⁣⋯ ⁣4244\!\cdots\!42T2+ T^{2} + 39 ⁣⋯ ⁣2839\!\cdots\!28p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
71D4D_{4} 1+ 1 + 24 ⁣⋯ ⁣7624\!\cdots\!76T+ T + 13 ⁣⋯ ⁣2613\!\cdots\!26T2+ T^{2} + 24 ⁣⋯ ⁣7624\!\cdots\!76p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
73D4D_{4} 1+ 1 + 12 ⁣⋯ ⁣9612\!\cdots\!96T+ T + 20 ⁣⋯ ⁣9820\!\cdots\!98T2+ T^{2} + 12 ⁣⋯ ⁣9612\!\cdots\!96p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
79D4D_{4} 1− 1 - 14 ⁣⋯ ⁣8014\!\cdots\!80T+ T + 34 ⁣⋯ ⁣1834\!\cdots\!18T2− T^{2} - 14 ⁣⋯ ⁣8014\!\cdots\!80p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
83D4D_{4} 1− 1 - 11 ⁣⋯ ⁣8411\!\cdots\!84T+ T + 15 ⁣⋯ ⁣1815\!\cdots\!18T2− T^{2} - 11 ⁣⋯ ⁣8411\!\cdots\!84p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
89D4D_{4} 1− 1 - 13 ⁣⋯ ⁣2013\!\cdots\!20T+ T + 29 ⁣⋯ ⁣5829\!\cdots\!58T2− T^{2} - 13 ⁣⋯ ⁣2013\!\cdots\!20p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
97D4D_{4} 1− 1 - 11 ⁣⋯ ⁣9211\!\cdots\!92T+ T + 11 ⁣⋯ ⁣4211\!\cdots\!42T2− T^{2} - 11 ⁣⋯ ⁣9211\!\cdots\!92p27T3+p54T4 p^{27} T^{3} + p^{54} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−14.06477204099351232904679139845, −13.38410926905362476763089848329, −12.27102297236651670439491415361, −12.07537324855380251712478265333, −11.39646639776216202051138602218, −11.37692912392080901369948708456, −10.08278243994427378294633847993, −9.429053755919024524212480593263, −7.81429847586326825084697725455, −7.27043702409289217151549939667, −6.32825728213086568247299599395, −6.12260802869509641084914037188, −5.06964209794475153978336899621, −4.70355361070169436196491332199, −3.78658085621396615232428513966, −3.39663315239867535633678402502, −1.98621832638906211321755447021, −1.32546194362356864904919110011, 0, 0, 1.32546194362356864904919110011, 1.98621832638906211321755447021, 3.39663315239867535633678402502, 3.78658085621396615232428513966, 4.70355361070169436196491332199, 5.06964209794475153978336899621, 6.12260802869509641084914037188, 6.32825728213086568247299599395, 7.27043702409289217151549939667, 7.81429847586326825084697725455, 9.429053755919024524212480593263, 10.08278243994427378294633847993, 11.37692912392080901369948708456, 11.39646639776216202051138602218, 12.07537324855380251712478265333, 12.27102297236651670439491415361, 13.38410926905362476763089848329, 14.06477204099351232904679139845

Graph of the ZZ-function along the critical line