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Properties

Label 2-87e2-1.1-c1-0-53
Degree 22
Conductor 75697569
Sign 11
Analytic cond. 60.438760.4387
Root an. cond. 7.774237.77423
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.09·2-s + 2.37·4-s − 4.22·5-s − 1.68·7-s − 0.783·8-s + 8.83·10-s − 2.06·11-s + 4.77·13-s + 3.52·14-s − 3.11·16-s + 0.783·17-s + 0.328·19-s − 10.0·20-s + 4.32·22-s + 9.05·23-s + 12.8·25-s − 9.98·26-s − 3.99·28-s + 0.158·31-s + 8.07·32-s − 1.63·34-s + 7.11·35-s − 1.11·37-s − 0.688·38-s + 3.30·40-s + 9.30·41-s − 1.00·43-s + ⋯
L(s)  = 1  − 1.47·2-s + 1.18·4-s − 1.88·5-s − 0.636·7-s − 0.276·8-s + 2.79·10-s − 0.622·11-s + 1.32·13-s + 0.941·14-s − 0.777·16-s + 0.189·17-s + 0.0754·19-s − 2.24·20-s + 0.921·22-s + 1.88·23-s + 2.56·25-s − 1.95·26-s − 0.755·28-s + 0.0283·31-s + 1.42·32-s − 0.280·34-s + 1.20·35-s − 0.182·37-s − 0.111·38-s + 0.523·40-s + 1.45·41-s − 0.153·43-s + ⋯

Functional equation

Λ(s)=(7569s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(7569s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 75697569    =    32⋅2923^{2} \cdot 29^{2}
Sign: 11
Analytic conductor: 60.438760.4387
Root analytic conductor: 7.774237.77423
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 7569, ( :1/2), 1)(2,\ 7569,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.48153055780.4815305578
L(12)L(\frac12) ≈\approx 0.48153055780.4815305578
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}
ppFp(T)F_p(T)
bad3 1 1
29 1 1
good2 1+2.09T+2T2 1 + 2.09T + 2T^{2}
5 1+4.22T+5T2 1 + 4.22T + 5T^{2}
7 1+1.68T+7T2 1 + 1.68T + 7T^{2}
11 1+2.06T+11T2 1 + 2.06T + 11T^{2}
13 1−4.77T+13T2 1 - 4.77T + 13T^{2}
17 1−0.783T+17T2 1 - 0.783T + 17T^{2}
19 1−0.328T+19T2 1 - 0.328T + 19T^{2}
23 1−9.05T+23T2 1 - 9.05T + 23T^{2}
31 1−0.158T+31T2 1 - 0.158T + 31T^{2}
37 1+1.11T+37T2 1 + 1.11T + 37T^{2}
41 1−9.30T+41T2 1 - 9.30T + 41T^{2}
43 1+1.00T+43T2 1 + 1.00T + 43T^{2}
47 1−10.5T+47T2 1 - 10.5T + 47T^{2}
53 1+5.39T+53T2 1 + 5.39T + 53T^{2}
59 1+11.2T+59T2 1 + 11.2T + 59T^{2}
61 1−2.11T+61T2 1 - 2.11T + 61T^{2}
67 1−4.98T+67T2 1 - 4.98T + 67T^{2}
71 1+1.82T+71T2 1 + 1.82T + 71T^{2}
73 1−11.6T+73T2 1 - 11.6T + 73T^{2}
79 1+8.47T+79T2 1 + 8.47T + 79T^{2}
83 1−2.46T+83T2 1 - 2.46T + 83T^{2}
89 1+10.0T+89T2 1 + 10.0T + 89T^{2}
97 1+4.67T+97T2 1 + 4.67T + 97T^{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

−8.009527937789017319477025086495, −7.36509134172736299869489217754, −6.96291959543119620363763736922, −6.10678503368465372643847020250, −4.96200740904113207734406063449, −4.18106655826067159313173923705, −3.38099362895615812340330467427, −2.74340347721595922088946699742, −1.20667867883353681301510065083, −0.51335314235516420149401676383, 0.51335314235516420149401676383, 1.20667867883353681301510065083, 2.74340347721595922088946699742, 3.38099362895615812340330467427, 4.18106655826067159313173923705, 4.96200740904113207734406063449, 6.10678503368465372643847020250, 6.96291959543119620363763736922, 7.36509134172736299869489217754, 8.009527937789017319477025086495

Graph of the ZZ-function along the critical line