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Properties

Label 2-6525-1.1-c1-0-10
Degree 22
Conductor 65256525
Sign 11
Analytic cond. 52.102352.1023
Root an. cond. 7.218197.21819
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  − 1.93·2-s + 1.74·4-s − 3.68·7-s + 0.491·8-s + 0.318·11-s − 4.18·13-s + 7.12·14-s − 4.44·16-s + 3.17·17-s − 5.87·19-s − 0.616·22-s + 2.50·23-s + 8.10·26-s − 6.42·28-s − 29-s + 2.50·31-s + 7.61·32-s − 6.14·34-s − 7.87·37-s + 11.3·38-s − 8.72·41-s + 10.7·43-s + 0.556·44-s − 4.85·46-s − 11.0·47-s + 6.55·49-s − 7.31·52-s + ⋯
L(s)  = 1  − 1.36·2-s + 0.872·4-s − 1.39·7-s + 0.173·8-s + 0.0960·11-s − 1.16·13-s + 1.90·14-s − 1.11·16-s + 0.769·17-s − 1.34·19-s − 0.131·22-s + 0.522·23-s + 1.59·26-s − 1.21·28-s − 0.185·29-s + 0.450·31-s + 1.34·32-s − 1.05·34-s − 1.29·37-s + 1.84·38-s − 1.36·41-s + 1.63·43-s + 0.0838·44-s − 0.715·46-s − 1.61·47-s + 0.936·49-s − 1.01·52-s + ⋯

Functional equation

Λ(s)=(6525s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6525s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 6525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 65256525    =    32⋅52⋅293^{2} \cdot 5^{2} \cdot 29
Sign: 11
Analytic conductor: 52.102352.1023
Root analytic conductor: 7.218197.21819
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6525, ( :1/2), 1)(2,\ 6525,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 0.28491572580.2849157258
L(12)L(\frac12) ≈\approx 0.28491572580.2849157258
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
5 1 1
29 1+T 1 + T
good2 1+1.93T+2T2 1 + 1.93T + 2T^{2}
7 1+3.68T+7T2 1 + 3.68T + 7T^{2}
11 1−0.318T+11T2 1 - 0.318T + 11T^{2}
13 1+4.18T+13T2 1 + 4.18T + 13T^{2}
17 1−3.17T+17T2 1 - 3.17T + 17T^{2}
19 1+5.87T+19T2 1 + 5.87T + 19T^{2}
23 1−2.50T+23T2 1 - 2.50T + 23T^{2}
31 1−2.50T+31T2 1 - 2.50T + 31T^{2}
37 1+7.87T+37T2 1 + 7.87T + 37T^{2}
41 1+8.72T+41T2 1 + 8.72T + 41T^{2}
43 1−10.7T+43T2 1 - 10.7T + 43T^{2}
47 1+11.0T+47T2 1 + 11.0T + 47T^{2}
53 1+8.24T+53T2 1 + 8.24T + 53T^{2}
59 1−11.3T+59T2 1 - 11.3T + 59T^{2}
61 1+3.87T+61T2 1 + 3.87T + 61T^{2}
67 1+7.04T+67T2 1 + 7.04T + 67T^{2}
71 1+6.24T+71T2 1 + 6.24T + 71T^{2}
73 1−7.87T+73T2 1 - 7.87T + 73T^{2}
79 1+4.85T+79T2 1 + 4.85T + 79T^{2}
83 1+8.37T+83T2 1 + 8.37T + 83T^{2}
89 1−15.9T+89T2 1 - 15.9T + 89T^{2}
97 1+11.2T+97T2 1 + 11.2T + 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

−8.173385257084861133125841336586, −7.32589819081711650475092931469, −6.84216376256346557295278237747, −6.24306926658412308380205356443, −5.21906591939883015519204068482, −4.37922029924548871488388212833, −3.37216249108637899689435483168, −2.57561499579602213445991218715, −1.61003489845502388740085935893, −0.33560787579795909468341585224, 0.33560787579795909468341585224, 1.61003489845502388740085935893, 2.57561499579602213445991218715, 3.37216249108637899689435483168, 4.37922029924548871488388212833, 5.21906591939883015519204068482, 6.24306926658412308380205356443, 6.84216376256346557295278237747, 7.32589819081711650475092931469, 8.173385257084861133125841336586

Graph of the ZZ-function along the critical line