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Properties

Label 2-2175-1.1-c1-0-51
Degree 22
Conductor 21752175
Sign −1-1
Analytic cond. 17.367417.3674
Root an. cond. 4.167424.16742
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.51·2-s − 3-s + 4.34·4-s + 2.51·6-s − 0.173·7-s − 5.90·8-s + 9-s + 5.08·11-s − 4.34·12-s − 1.82·13-s + 0.436·14-s + 6.19·16-s + 4.24·17-s − 2.51·18-s − 8.62·19-s + 0.173·21-s − 12.8·22-s − 3.16·23-s + 5.90·24-s + 4.60·26-s − 27-s − 0.753·28-s + 29-s + 3.22·31-s − 3.78·32-s − 5.08·33-s − 10.6·34-s + ⋯
L(s)  = 1  − 1.78·2-s − 0.577·3-s + 2.17·4-s + 1.02·6-s − 0.0655·7-s − 2.08·8-s + 0.333·9-s + 1.53·11-s − 1.25·12-s − 0.506·13-s + 0.116·14-s + 1.54·16-s + 1.02·17-s − 0.593·18-s − 1.97·19-s + 0.0378·21-s − 2.72·22-s − 0.659·23-s + 1.20·24-s + 0.902·26-s − 0.192·27-s − 0.142·28-s + 0.185·29-s + 0.578·31-s − 0.669·32-s − 0.884·33-s − 1.83·34-s + ⋯

Functional equation

Λ(s)=(2175s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(2175s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 21752175    =    3⋅52⋅293 \cdot 5^{2} \cdot 29
Sign: −1-1
Analytic conductor: 17.367417.3674
Root analytic conductor: 4.167424.16742
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 2175, ( :1/2), −1)(2,\ 2175,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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}
ppFp(T)F_p(T)
bad3 1+T 1 + T
5 1 1
29 1−T 1 - T
good2 1+2.51T+2T2 1 + 2.51T + 2T^{2}
7 1+0.173T+7T2 1 + 0.173T + 7T^{2}
11 1−5.08T+11T2 1 - 5.08T + 11T^{2}
13 1+1.82T+13T2 1 + 1.82T + 13T^{2}
17 1−4.24T+17T2 1 - 4.24T + 17T^{2}
19 1+8.62T+19T2 1 + 8.62T + 19T^{2}
23 1+3.16T+23T2 1 + 3.16T + 23T^{2}
31 1−3.22T+31T2 1 - 3.22T + 31T^{2}
37 1+1.97T+37T2 1 + 1.97T + 37T^{2}
41 1+9.96T+41T2 1 + 9.96T + 41T^{2}
43 1+7.91T+43T2 1 + 7.91T + 43T^{2}
47 1−8.66T+47T2 1 - 8.66T + 47T^{2}
53 1+5.40T+53T2 1 + 5.40T + 53T^{2}
59 1−7.66T+59T2 1 - 7.66T + 59T^{2}
61 1−2.76T+61T2 1 - 2.76T + 61T^{2}
67 1+8.13T+67T2 1 + 8.13T + 67T^{2}
71 1+6.25T+71T2 1 + 6.25T + 71T^{2}
73 1−6.74T+73T2 1 - 6.74T + 73T^{2}
79 1−4.54T+79T2 1 - 4.54T + 79T^{2}
83 1−11.6T+83T2 1 - 11.6T + 83T^{2}
89 1+16.4T+89T2 1 + 16.4T + 89T^{2}
97 1+2.74T+97T2 1 + 2.74T + 97T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−8.634289428123115981019371274873, −8.188114274881213956221648721078, −7.14982176866421071439925990887, −6.57881900002486994031333451663, −6.01355311778597859270131890359, −4.67338120914734568974097030334, −3.56343859013510770513136402419, −2.14254364341070777381892870074, −1.28165043048159175423304649322, 0, 1.28165043048159175423304649322, 2.14254364341070777381892870074, 3.56343859013510770513136402419, 4.67338120914734568974097030334, 6.01355311778597859270131890359, 6.57881900002486994031333451663, 7.14982176866421071439925990887, 8.188114274881213956221648721078, 8.634289428123115981019371274873

Graph of the ZZ-function along the critical line