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Properties

Label 2-6525-1.1-c1-0-22
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  + 0.200·2-s − 1.95·4-s − 0.775·7-s − 0.794·8-s − 4.45·11-s − 3.67·13-s − 0.155·14-s + 3.76·16-s + 1.05·17-s + 3.76·19-s − 0.893·22-s + 6.21·23-s − 0.738·26-s + 1.51·28-s − 29-s − 3.96·31-s + 2.34·32-s + 0.211·34-s − 2.64·37-s + 0.754·38-s − 6.67·41-s − 9.96·43-s + 8.72·44-s + 1.24·46-s − 12.6·47-s − 6.39·49-s + 7.21·52-s + ⋯
L(s)  = 1  + 0.141·2-s − 0.979·4-s − 0.292·7-s − 0.280·8-s − 1.34·11-s − 1.02·13-s − 0.0415·14-s + 0.940·16-s + 0.256·17-s + 0.863·19-s − 0.190·22-s + 1.29·23-s − 0.144·26-s + 0.287·28-s − 0.185·29-s − 0.711·31-s + 0.414·32-s + 0.0363·34-s − 0.434·37-s + 0.122·38-s − 1.04·41-s − 1.51·43-s + 1.31·44-s + 0.183·46-s − 1.84·47-s − 0.914·49-s + 0.999·52-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 0.79612888180.7961288818
L(12)L(\frac12) ≈\approx 0.79612888180.7961288818
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 1−0.200T+2T2 1 - 0.200T + 2T^{2}
7 1+0.775T+7T2 1 + 0.775T + 7T^{2}
11 1+4.45T+11T2 1 + 4.45T + 11T^{2}
13 1+3.67T+13T2 1 + 3.67T + 13T^{2}
17 1−1.05T+17T2 1 - 1.05T + 17T^{2}
19 1−3.76T+19T2 1 - 3.76T + 19T^{2}
23 1−6.21T+23T2 1 - 6.21T + 23T^{2}
31 1+3.96T+31T2 1 + 3.96T + 31T^{2}
37 1+2.64T+37T2 1 + 2.64T + 37T^{2}
41 1+6.67T+41T2 1 + 6.67T + 41T^{2}
43 1+9.96T+43T2 1 + 9.96T + 43T^{2}
47 1+12.6T+47T2 1 + 12.6T + 47T^{2}
53 1+2.87T+53T2 1 + 2.87T + 53T^{2}
59 1−7.92T+59T2 1 - 7.92T + 59T^{2}
61 1+2.22T+61T2 1 + 2.22T + 61T^{2}
67 1−4.12T+67T2 1 - 4.12T + 67T^{2}
71 1+13.6T+71T2 1 + 13.6T + 71T^{2}
73 1−8.60T+73T2 1 - 8.60T + 73T^{2}
79 1−6.97T+79T2 1 - 6.97T + 79T^{2}
83 1−2.90T+83T2 1 - 2.90T + 83T^{2}
89 1−4.30T+89T2 1 - 4.30T + 89T^{2}
97 1+3.82T+97T2 1 + 3.82T + 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

−8.056519734745020664080131470398, −7.37167576315954373378703927133, −6.68703198374115160085607976101, −5.56565702576643013528397017259, −5.06887066294013350887717148382, −4.71301940956515197924240048806, −3.34213468967141299398206542467, −3.12150596212515588374759967328, −1.82276137764036478220528016386, −0.44169320812110696027517238890, 0.44169320812110696027517238890, 1.82276137764036478220528016386, 3.12150596212515588374759967328, 3.34213468967141299398206542467, 4.71301940956515197924240048806, 5.06887066294013350887717148382, 5.56565702576643013528397017259, 6.68703198374115160085607976101, 7.37167576315954373378703927133, 8.056519734745020664080131470398

Graph of the ZZ-function along the critical line