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Properties

Label 4-987696-1.1-c1e2-0-10
Degree 44
Conductor 987696987696
Sign −1-1
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 11

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: −1-1
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: 11
Selberg data: (4, 987696, ( :1/2,1/2), −1)(4,\ 987696,\ (\ :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)
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−T+pT2)(1+8T+pT2) ( 1 - T + p T^{2} )( 1 + 8 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+3T+pT2)(1+9T+pT2) ( 1 + 3 T + p T^{2} )( 1 + 9 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−T+pT2)(1+2T+pT2) ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} )
71C2C_2×\timesC2C_2 (1−3T+pT2)(1+12T+pT2) ( 1 - 3 T + p T^{2} )( 1 + 12 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

−7.81947458032215673314917532689, −7.49134181637822409019182211683, −6.90820799466782018221444454712, −6.51538629427949879427875269097, −6.08900451382462637094878921273, −5.66001524938879741637465838582, −5.36968156418594919959726385528, −4.88323745463589915070432771723, −4.46103397436716246743238818421, −3.82221976196370896553630440730, −2.99753924749007578976059399541, −2.09439797072401209883470616174, −1.76289656293079236896734150247, −1.12946967656516647051616965624, 0, 1.12946967656516647051616965624, 1.76289656293079236896734150247, 2.09439797072401209883470616174, 2.99753924749007578976059399541, 3.82221976196370896553630440730, 4.46103397436716246743238818421, 4.88323745463589915070432771723, 5.36968156418594919959726385528, 5.66001524938879741637465838582, 6.08900451382462637094878921273, 6.51538629427949879427875269097, 6.90820799466782018221444454712, 7.49134181637822409019182211683, 7.81947458032215673314917532689

Graph of the ZZ-function along the critical line