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Properties

Label 2-87e2-1.1-c1-0-61
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  − 1.13·2-s − 0.714·4-s + 0.407·5-s + 2.15·7-s + 3.07·8-s − 0.461·10-s − 3.55·11-s − 0.0966·13-s − 2.44·14-s − 2.05·16-s − 0.587·17-s + 3.61·19-s − 0.291·20-s + 4.02·22-s − 8.26·23-s − 4.83·25-s + 0.109·26-s − 1.54·28-s − 1.67·31-s − 3.82·32-s + 0.665·34-s + 0.878·35-s + 0.0122·37-s − 4.10·38-s + 1.25·40-s + 9.57·41-s − 3.47·43-s + ⋯
L(s)  = 1  − 0.801·2-s − 0.357·4-s + 0.182·5-s + 0.815·7-s + 1.08·8-s − 0.145·10-s − 1.07·11-s − 0.0268·13-s − 0.653·14-s − 0.514·16-s − 0.142·17-s + 0.830·19-s − 0.0650·20-s + 0.858·22-s − 1.72·23-s − 0.966·25-s + 0.0214·26-s − 0.291·28-s − 0.301·31-s − 0.675·32-s + 0.114·34-s + 0.148·35-s + 0.00202·37-s − 0.665·38-s + 0.198·40-s + 1.49·41-s − 0.530·43-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 0.91629891050.9162989105
L(12)L(\frac12) ≈\approx 0.91629891050.9162989105
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+1.13T+2T2 1 + 1.13T + 2T^{2}
5 1−0.407T+5T2 1 - 0.407T + 5T^{2}
7 1−2.15T+7T2 1 - 2.15T + 7T^{2}
11 1+3.55T+11T2 1 + 3.55T + 11T^{2}
13 1+0.0966T+13T2 1 + 0.0966T + 13T^{2}
17 1+0.587T+17T2 1 + 0.587T + 17T^{2}
19 1−3.61T+19T2 1 - 3.61T + 19T^{2}
23 1+8.26T+23T2 1 + 8.26T + 23T^{2}
31 1+1.67T+31T2 1 + 1.67T + 31T^{2}
37 1−0.0122T+37T2 1 - 0.0122T + 37T^{2}
41 1−9.57T+41T2 1 - 9.57T + 41T^{2}
43 1+3.47T+43T2 1 + 3.47T + 43T^{2}
47 1+3.31T+47T2 1 + 3.31T + 47T^{2}
53 1+6.88T+53T2 1 + 6.88T + 53T^{2}
59 1−6.45T+59T2 1 - 6.45T + 59T^{2}
61 1−9.67T+61T2 1 - 9.67T + 61T^{2}
67 1−8.02T+67T2 1 - 8.02T + 67T^{2}
71 1−9.30T+71T2 1 - 9.30T + 71T^{2}
73 1−9.28T+73T2 1 - 9.28T + 73T^{2}
79 1−11.0T+79T2 1 - 11.0T + 79T^{2}
83 1+12.8T+83T2 1 + 12.8T + 83T^{2}
89 1+2.23T+89T2 1 + 2.23T + 89T^{2}
97 1+5.72T+97T2 1 + 5.72T + 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

−7.962804554713970590171740883939, −7.62527128958780248315524806817, −6.63273018605150687646677304493, −5.60233745632645068824528821067, −5.17857814466151424707271668323, −4.35810318471914091253424672550, −3.62992164238184430924244322196, −2.35267056768649447646175877615, −1.70924272134575210928263780891, −0.54797940048731320516837461728, 0.54797940048731320516837461728, 1.70924272134575210928263780891, 2.35267056768649447646175877615, 3.62992164238184430924244322196, 4.35810318471914091253424672550, 5.17857814466151424707271668323, 5.60233745632645068824528821067, 6.63273018605150687646677304493, 7.62527128958780248315524806817, 7.962804554713970590171740883939

Graph of the ZZ-function along the critical line