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Properties

Label 2-1575-1.1-c3-0-136
Degree 22
Conductor 15751575
Sign −1-1
Analytic cond. 92.928092.9280
Root an. cond. 9.639919.63991
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.84·2-s + 15.4·4-s − 7·7-s + 36.0·8-s − 62.1·11-s − 14.0·13-s − 33.8·14-s + 50.9·16-s + 63.5·17-s + 48.7·19-s − 301.·22-s − 99.3·23-s − 68.2·26-s − 108.·28-s + 69.0·29-s − 9.68·31-s − 41.7·32-s + 307.·34-s − 240.·37-s + 235.·38-s − 335.·41-s − 51.2·43-s − 960.·44-s − 481.·46-s − 451.·47-s + 49·49-s − 217.·52-s + ⋯
L(s)  = 1  + 1.71·2-s + 1.93·4-s − 0.377·7-s + 1.59·8-s − 1.70·11-s − 0.300·13-s − 0.646·14-s + 0.795·16-s + 0.906·17-s + 0.588·19-s − 2.91·22-s − 0.900·23-s − 0.514·26-s − 0.729·28-s + 0.442·29-s − 0.0561·31-s − 0.230·32-s + 1.55·34-s − 1.06·37-s + 1.00·38-s − 1.27·41-s − 0.181·43-s − 3.29·44-s − 1.54·46-s − 1.40·47-s + 0.142·49-s − 0.580·52-s + ⋯

Functional equation

Λ(s)=(1575s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(1575s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15751575    =    32⋅52⋅73^{2} \cdot 5^{2} \cdot 7
Sign: −1-1
Analytic conductor: 92.928092.9280
Root analytic conductor: 9.639919.63991
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1575, ( :3/2), −1)(2,\ 1575,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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)
bad3 1 1
5 1 1
7 1+7T 1 + 7T
good2 1−4.84T+8T2 1 - 4.84T + 8T^{2}
11 1+62.1T+1.33e3T2 1 + 62.1T + 1.33e3T^{2}
13 1+14.0T+2.19e3T2 1 + 14.0T + 2.19e3T^{2}
17 1−63.5T+4.91e3T2 1 - 63.5T + 4.91e3T^{2}
19 1−48.7T+6.85e3T2 1 - 48.7T + 6.85e3T^{2}
23 1+99.3T+1.21e4T2 1 + 99.3T + 1.21e4T^{2}
29 1−69.0T+2.43e4T2 1 - 69.0T + 2.43e4T^{2}
31 1+9.68T+2.97e4T2 1 + 9.68T + 2.97e4T^{2}
37 1+240.T+5.06e4T2 1 + 240.T + 5.06e4T^{2}
41 1+335.T+6.89e4T2 1 + 335.T + 6.89e4T^{2}
43 1+51.2T+7.95e4T2 1 + 51.2T + 7.95e4T^{2}
47 1+451.T+1.03e5T2 1 + 451.T + 1.03e5T^{2}
53 1+180.T+1.48e5T2 1 + 180.T + 1.48e5T^{2}
59 1+268.T+2.05e5T2 1 + 268.T + 2.05e5T^{2}
61 1+323.T+2.26e5T2 1 + 323.T + 2.26e5T^{2}
67 1−541.T+3.00e5T2 1 - 541.T + 3.00e5T^{2}
71 1−161.T+3.57e5T2 1 - 161.T + 3.57e5T^{2}
73 1+305.T+3.89e5T2 1 + 305.T + 3.89e5T^{2}
79 1+504.T+4.93e5T2 1 + 504.T + 4.93e5T^{2}
83 1+513.T+5.71e5T2 1 + 513.T + 5.71e5T^{2}
89 1+543.T+7.04e5T2 1 + 543.T + 7.04e5T^{2}
97 1+1.86e3T+9.12e5T2 1 + 1.86e3T + 9.12e5T^{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

−8.384331824567338309409869433142, −7.60758832949921062586527399256, −6.83522105699567632849114607417, −5.87399555005394543990128888500, −5.27513364180265903058296641855, −4.63451614880122821168500082308, −3.42287669883448047211874415962, −2.93597061187632528117779620375, −1.85440271616855420699768383103, 0, 1.85440271616855420699768383103, 2.93597061187632528117779620375, 3.42287669883448047211874415962, 4.63451614880122821168500082308, 5.27513364180265903058296641855, 5.87399555005394543990128888500, 6.83522105699567632849114607417, 7.60758832949921062586527399256, 8.384331824567338309409869433142

Graph of the ZZ-function along the critical line