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Properties

Label 2-350-1.1-c3-0-18
Degree 22
Conductor 350350
Sign 11
Analytic cond. 20.650620.6506
Root an. cond. 4.544304.54430
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 8·3-s + 4·4-s + 16·6-s + 7·7-s + 8·8-s + 37·9-s + 68·11-s + 32·12-s − 34·13-s + 14·14-s + 16·16-s − 74·17-s + 74·18-s − 128·19-s + 56·21-s + 136·22-s + 80·23-s + 64·24-s − 68·26-s + 80·27-s + 28·28-s + 286·29-s − 24·31-s + 32·32-s + 544·33-s − 148·34-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.53·3-s + 1/2·4-s + 1.08·6-s + 0.377·7-s + 0.353·8-s + 1.37·9-s + 1.86·11-s + 0.769·12-s − 0.725·13-s + 0.267·14-s + 1/4·16-s − 1.05·17-s + 0.968·18-s − 1.54·19-s + 0.581·21-s + 1.31·22-s + 0.725·23-s + 0.544·24-s − 0.512·26-s + 0.570·27-s + 0.188·28-s + 1.83·29-s − 0.139·31-s + 0.176·32-s + 2.86·33-s − 0.746·34-s + ⋯

Functional equation

Λ(s)=(350s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(350s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 350350    =    2⋅52⋅72 \cdot 5^{2} \cdot 7
Sign: 11
Analytic conductor: 20.650620.6506
Root analytic conductor: 4.544304.54430
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 350, ( :3/2), 1)(2,\ 350,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 5.2114804545.211480454
L(12)L(\frac12) ≈\approx 5.2114804545.211480454
L(52)L(\frac{5}{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)
bad2 1−pT 1 - p T
5 1 1
7 1−pT 1 - p T
good3 1−8T+p3T2 1 - 8 T + p^{3} T^{2}
11 1−68T+p3T2 1 - 68 T + p^{3} T^{2}
13 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
17 1+74T+p3T2 1 + 74 T + p^{3} T^{2}
19 1+128T+p3T2 1 + 128 T + p^{3} T^{2}
23 1−80T+p3T2 1 - 80 T + p^{3} T^{2}
29 1−286T+p3T2 1 - 286 T + p^{3} T^{2}
31 1+24T+p3T2 1 + 24 T + p^{3} T^{2}
37 1+294T+p3T2 1 + 294 T + p^{3} T^{2}
41 1−66T+p3T2 1 - 66 T + p^{3} T^{2}
43 1−124T+p3T2 1 - 124 T + p^{3} T^{2}
47 1+312T+p3T2 1 + 312 T + p^{3} T^{2}
53 1−34T+p3T2 1 - 34 T + p^{3} T^{2}
59 1−168T+p3T2 1 - 168 T + p^{3} T^{2}
61 1−170T+p3T2 1 - 170 T + p^{3} T^{2}
67 1+564T+p3T2 1 + 564 T + p^{3} T^{2}
71 1−616T+p3T2 1 - 616 T + p^{3} T^{2}
73 1+250T+p3T2 1 + 250 T + p^{3} T^{2}
79 1+944T+p3T2 1 + 944 T + p^{3} T^{2}
83 1+672T+p3T2 1 + 672 T + p^{3} T^{2}
89 1+1430T+p3T2 1 + 1430 T + p^{3} T^{2}
97 1−1270T+p3T2 1 - 1270 T + p^{3} T^{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

−11.18955576209801152537373407883, −10.02646495774156413541411468570, −8.899232880691086136442173117246, −8.500200986133607446452526270200, −7.13494435573915775696510398238, −6.45702957381836343965251844849, −4.64206236789129304092287574140, −3.94117041102020446291137735244, −2.71686229643575911675709249989, −1.67080333129789026443695544768, 1.67080333129789026443695544768, 2.71686229643575911675709249989, 3.94117041102020446291137735244, 4.64206236789129304092287574140, 6.45702957381836343965251844849, 7.13494435573915775696510398238, 8.500200986133607446452526270200, 8.899232880691086136442173117246, 10.02646495774156413541411468570, 11.18955576209801152537373407883

Graph of the ZZ-function along the critical line