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Properties

Label 4-987696-1.1-c1e2-0-6
Degree 44
Conductor 987696987696
Sign 11
Analytic cond. 62.976362.9763
Root an. cond. 2.817042.81704
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s − 4-s + 5·5-s + 2·6-s − 3·8-s + 3·9-s + 5·10-s − 2·12-s + 10·15-s − 16-s − 3·17-s + 3·18-s − 19-s − 5·20-s − 6·24-s + 11·25-s + 4·27-s + 10·30-s + 7·31-s + 5·32-s − 3·34-s − 3·36-s − 38-s − 15·40-s + 15·45-s − 2·48-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 1/2·4-s + 2.23·5-s + 0.816·6-s − 1.06·8-s + 9-s + 1.58·10-s − 0.577·12-s + 2.58·15-s − 1/4·16-s − 0.727·17-s + 0.707·18-s − 0.229·19-s − 1.11·20-s − 1.22·24-s + 11/5·25-s + 0.769·27-s + 1.82·30-s + 1.25·31-s + 0.883·32-s − 0.514·34-s − 1/2·36-s − 0.162·38-s − 2.37·40-s + 2.23·45-s − 0.288·48-s + ⋯

Functional equation

Λ(s)=(987696s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(987696s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 987696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 987696987696    =    24⋅32⋅1932^{4} \cdot 3^{2} \cdot 19^{3}
Sign: 11
Analytic conductor: 62.976362.9763
Root analytic conductor: 2.817042.81704
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 987696, ( :1/2,1/2), 1)(4,\ 987696,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.0405010536.040501053
L(12)L(\frac12) ≈\approx 6.0405010536.040501053
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)
bad2C2C_2 1−T+pT2 1 - T + p T^{2}
3C1C_1 (1−T)2 ( 1 - T )^{2}
19C1C_1 1+T 1 + T
good5C2C_2×\timesC2C_2 (1−4T+pT2)(1−T+pT2) ( 1 - 4 T + p T^{2} )( 1 - T + p T^{2} )
7C2C_2 (1−3T+pT2)(1+3T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} )
11C2C_2 (1−6T+pT2)(1+6T+pT2) ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} )
13C22C_2^2 1+8T2+p2T4 1 + 8 T^{2} + p^{2} T^{4}
17C2C_2×\timesC2C_2 (1+pT2)(1+3T+pT2) ( 1 + p T^{2} )( 1 + 3 T + p T^{2} )
23C22C_2^2 1−24T2+p2T4 1 - 24 T^{2} + p^{2} T^{4}
29C22C_2^2 1+27T2+p2T4 1 + 27 T^{2} + p^{2} T^{4}
31C2C_2×\timesC2C_2 (1−8T+pT2)(1+T+pT2) ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} )
37C2C_2 (1−8T+pT2)(1+8T+pT2) ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} )
41C22C_2^2 1−3T2+p2T4 1 - 3 T^{2} + p^{2} T^{4}
43C22C_2^2 1+25T2+p2T4 1 + 25 T^{2} + p^{2} T^{4}
47C22C_2^2 1−68T2+p2T4 1 - 68 T^{2} + p^{2} T^{4}
53C22C_2^2 1+79T2+p2T4 1 + 79 T^{2} + p^{2} T^{4}
59C2C_2×\timesC2C_2 (1−9T+pT2)(1−3T+pT2) ( 1 - 9 T + p T^{2} )( 1 - 3 T + p T^{2} )
61C2C_2×\timesC2C_2 (1−10T+pT2)(1+5T+pT2) ( 1 - 10 T + p T^{2} )( 1 + 5 T + p T^{2} )
67C2C_2×\timesC2C_2 (1−2T+pT2)(1+T+pT2) ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} )
71C2C_2×\timesC2C_2 (1−12T+pT2)(1+3T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} )
73C2C_2×\timesC2C_2 (1−13T+pT2)(1+11T+pT2) ( 1 - 13 T + p T^{2} )( 1 + 11 T + p T^{2} )
79C2C_2 (1−12T+pT2)(1+12T+pT2) ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} )
83C22C_2^2 1−72T2+p2T4 1 - 72 T^{2} + p^{2} T^{4}
89C22C_2^2 1+165T2+p2T4 1 + 165 T^{2} + p^{2} T^{4}
97C22C_2^2 1+58T2+p2T4 1 + 58 T^{2} + p^{2} 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

−8.301276033902248886072544324981, −7.83767424717188548791787990887, −7.08539267885969964936271339550, −6.63184230370292439861888314224, −6.33789817811069174496896442851, −5.83086745494965605328944737782, −5.47317821006990102026865378357, −4.85591809358781683871283314618, −4.59548332478518888234100145371, −3.90451504588036497167163213578, −3.39968332980185008642977401955, −2.75886969069619919972935688817, −2.27508195481470203047502626025, −1.94786254432634666588022972622, −0.987798022315836519928665919699, 0.987798022315836519928665919699, 1.94786254432634666588022972622, 2.27508195481470203047502626025, 2.75886969069619919972935688817, 3.39968332980185008642977401955, 3.90451504588036497167163213578, 4.59548332478518888234100145371, 4.85591809358781683871283314618, 5.47317821006990102026865378357, 5.83086745494965605328944737782, 6.33789817811069174496896442851, 6.63184230370292439861888314224, 7.08539267885969964936271339550, 7.83767424717188548791787990887, 8.301276033902248886072544324981

Graph of the ZZ-function along the critical line