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Properties

Label 4-1216e2-1.1-c1e2-0-19
Degree 44
Conductor 14786561478656
Sign 11
Analytic cond. 94.280394.2803
Root an. cond. 3.116053.11605
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 6·7-s + 5·9-s + 6·17-s − 18·23-s + 10·25-s + 12·31-s + 12·41-s + 13·49-s + 30·63-s + 22·73-s − 24·79-s + 16·81-s − 16·97-s + 12·103-s − 24·113-s + 36·119-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 30·153-s + 157-s − 108·161-s + 163-s + ⋯
L(s)  = 1  + 2.26·7-s + 5/3·9-s + 1.45·17-s − 3.75·23-s + 2·25-s + 2.15·31-s + 1.87·41-s + 13/7·49-s + 3.77·63-s + 2.57·73-s − 2.70·79-s + 16/9·81-s − 1.62·97-s + 1.18·103-s − 2.25·113-s + 3.30·119-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 2.42·153-s + 0.0798·157-s − 8.51·161-s + 0.0783·163-s + ⋯

Functional equation

Λ(s)=(1478656s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1478656s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14786561478656    =    212⋅1922^{12} \cdot 19^{2}
Sign: 11
Analytic conductor: 94.280394.2803
Root analytic conductor: 3.116053.11605
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1478656, ( :1/2,1/2), 1)(4,\ 1478656,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.0977614694.097761469
L(12)L(\frac12) ≈\approx 4.0977614694.097761469
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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
19C2C_2 1+T2 1 + T^{2}
good3C22C_2^2 1−5T2+p2T4 1 - 5 T^{2} + p^{2} T^{4}
5C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
7C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
11C2C_2 (1−pT2)2 ( 1 - p T^{2} )^{2}
13C22C_2^2 1−17T2+p2T4 1 - 17 T^{2} + p^{2} T^{4}
17C2C_2 (1−3T+pT2)2 ( 1 - 3 T + p T^{2} )^{2}
23C2C_2 (1+9T+pT2)2 ( 1 + 9 T + p T^{2} )^{2}
29C22C_2^2 1+23T2+p2T4 1 + 23 T^{2} + p^{2} T^{4}
31C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
37C22C_2^2 1−38T2+p2T4 1 - 38 T^{2} + p^{2} T^{4}
41C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
43C22C_2^2 1−22T2+p2T4 1 - 22 T^{2} + p^{2} T^{4}
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C22C_2^2 1−25T2+p2T4 1 - 25 T^{2} + p^{2} T^{4}
59C22C_2^2 1−109T2+p2T4 1 - 109 T^{2} + p^{2} T^{4}
61C22C_2^2 1−86T2+p2T4 1 - 86 T^{2} + p^{2} T^{4}
67C22C_2^2 1−109T2+p2T4 1 - 109 T^{2} + p^{2} T^{4}
71C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
73C2C_2 (1−11T+pT2)2 ( 1 - 11 T + p T^{2} )^{2}
79C2C_2 (1+12T+pT2)2 ( 1 + 12 T + p T^{2} )^{2}
83C22C_2^2 1−130T2+p2T4 1 - 130 T^{2} + p^{2} T^{4}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C2C_2 (1+8T+pT2)2 ( 1 + 8 T + p T^{2} )^{2}
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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

−9.862685587425394053191256341060, −9.809603195376979946669488949759, −9.145816530441427860233828752262, −8.462742243307877688021409433914, −8.105497545734099352261462033095, −8.040319249155155768203095438866, −7.58651326697501713051468396478, −7.28668043003824428231687611588, −6.46359765609492926333934285601, −6.37254600139472960295505130365, −5.51248430504078833539322710571, −5.35132135944355142848853256403, −4.57082014948238083712548335703, −4.39078225807291656867764951514, −4.14829853802996598658054914664, −3.40579877635572868078992330615, −2.52003549943530161976396997584, −2.04295540264713895441864435695, −1.32624340253520185142858625608, −1.07113267627942141792377710412, 1.07113267627942141792377710412, 1.32624340253520185142858625608, 2.04295540264713895441864435695, 2.52003549943530161976396997584, 3.40579877635572868078992330615, 4.14829853802996598658054914664, 4.39078225807291656867764951514, 4.57082014948238083712548335703, 5.35132135944355142848853256403, 5.51248430504078833539322710571, 6.37254600139472960295505130365, 6.46359765609492926333934285601, 7.28668043003824428231687611588, 7.58651326697501713051468396478, 8.040319249155155768203095438866, 8.105497545734099352261462033095, 8.462742243307877688021409433914, 9.145816530441427860233828752262, 9.809603195376979946669488949759, 9.862685587425394053191256341060

Graph of the ZZ-function along the critical line