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Properties

Label 4-312e2-1.1-c1e2-0-36
Degree 44
Conductor 9734497344
Sign −1-1
Analytic cond. 6.206736.20673
Root an. cond. 1.578391.57839
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s − 5·13-s + 5·25-s − 12·37-s − 11·49-s − 61-s − 27·73-s + 9·81-s + 15·117-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 9-s − 1.38·13-s + 25-s − 1.97·37-s − 1.57·49-s − 0.128·61-s − 3.16·73-s + 81-s + 1.38·117-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 + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + ⋯

Functional equation

Λ(s)=(97344s/2 ΓC(s)2 L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(97344s/2 ΓC(s+1/2)2 L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 9734497344    =    26⋅32⋅1322^{6} \cdot 3^{2} \cdot 13^{2}
Sign: −1-1
Analytic conductor: 6.206736.20673
Root analytic conductor: 1.578391.57839
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (4, 97344, ( :1/2,1/2), −1)(4,\ 97344,\ (\ :1/2, 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}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3C2C_2 1+pT2 1 + p T^{2}
13C2C_2 1+5T+pT2 1 + 5 T + p T^{2}
good5C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
7C22C_2^2 1+11T2+p2T4 1 + 11 T^{2} + p^{2} T^{4}
11C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
17C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
19C22C_2^2 1+26T2+p2T4 1 + 26 T^{2} + p^{2} T^{4}
23C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
29C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
31C22C_2^2 1+59T2+p2T4 1 + 59 T^{2} + p^{2} T^{4}
37C2C_2 (1+T+pT2)(1+11T+pT2) ( 1 + T + p T^{2} )( 1 + 11 T + p T^{2} )
41C22C_2^2 1−pT2+p2T4 1 - p T^{2} + p^{2} T^{4}
43C2C_2 (1−5T+pT2)(1+5T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} )
47C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
53C22C_2^2 1+pT2+p2T4 1 + p T^{2} + p^{2} T^{4}
59C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
61C2C_2 (1−13T+pT2)(1+14T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} )
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+10T+pT2)(1+17T+pT2) ( 1 + 10 T + p T^{2} )( 1 + 17 T + p T^{2} )
79C2C_2 (1−17T+pT2)(1+17T+pT2) ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} )
83C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
89C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
97C2C_2 (1−19T+pT2)(1+19T+pT2) ( 1 - 19 T + p T^{2} )( 1 + 19 T + p T^{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

−9.285859569587139990848089363651, −8.799834474592663518446554011854, −8.496156759644029146258936652067, −7.83514328877578084059175606366, −7.33486208797466574796310183246, −6.87754188197318428321351486880, −6.30121732037613294121156526239, −5.68781277044688638857092297285, −5.02630362326021944380565763766, −4.82295902328672719631606073357, −3.88161160984559063672153387352, −3.08086443998836126308130636598, −2.65306725856461153155748041157, −1.65340611096489772271595902239, 0, 1.65340611096489772271595902239, 2.65306725856461153155748041157, 3.08086443998836126308130636598, 3.88161160984559063672153387352, 4.82295902328672719631606073357, 5.02630362326021944380565763766, 5.68781277044688638857092297285, 6.30121732037613294121156526239, 6.87754188197318428321351486880, 7.33486208797466574796310183246, 7.83514328877578084059175606366, 8.496156759644029146258936652067, 8.799834474592663518446554011854, 9.285859569587139990848089363651

Graph of the ZZ-function along the critical line