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Properties

Label 2-9610-1.1-c1-0-82
Degree 22
Conductor 96109610
Sign 11
Analytic cond. 76.736276.7362
Root an. cond. 8.759928.75992
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-s + 2.28·3-s + 4-s − 5-s − 2.28·6-s − 1.23·7-s − 8-s + 2.23·9-s + 10-s + 1.95·11-s + 2.28·12-s − 0.874·13-s + 1.23·14-s − 2.28·15-s + 16-s + 6.32·17-s − 2.23·18-s − 2.76·19-s − 20-s − 2.82·21-s − 1.95·22-s − 3.16·23-s − 2.28·24-s + 25-s + 0.874·26-s − 1.74·27-s − 1.23·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.32·3-s + 0.5·4-s − 0.447·5-s − 0.934·6-s − 0.467·7-s − 0.353·8-s + 0.745·9-s + 0.316·10-s + 0.589·11-s + 0.660·12-s − 0.242·13-s + 0.330·14-s − 0.590·15-s + 0.250·16-s + 1.53·17-s − 0.527·18-s − 0.634·19-s − 0.223·20-s − 0.617·21-s − 0.416·22-s − 0.659·23-s − 0.467·24-s + 0.200·25-s + 0.171·26-s − 0.336·27-s − 0.233·28-s + ⋯

Functional equation

Λ(s)=(9610s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(9610s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9610 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 96109610    =    2⋅5⋅3122 \cdot 5 \cdot 31^{2}
Sign: 11
Analytic conductor: 76.736276.7362
Root analytic conductor: 8.759928.75992
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 9610, ( :1/2), 1)(2,\ 9610,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.1046242232.104624223
L(12)L(\frac12) ≈\approx 2.1046242232.104624223
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)
bad2 1+T 1 + T
5 1+T 1 + T
31 1 1
good3 1−2.28T+3T2 1 - 2.28T + 3T^{2}
7 1+1.23T+7T2 1 + 1.23T + 7T^{2}
11 1−1.95T+11T2 1 - 1.95T + 11T^{2}
13 1+0.874T+13T2 1 + 0.874T + 13T^{2}
17 1−6.32T+17T2 1 - 6.32T + 17T^{2}
19 1+2.76T+19T2 1 + 2.76T + 19T^{2}
23 1+3.16T+23T2 1 + 3.16T + 23T^{2}
29 1−6.86T+29T2 1 - 6.86T + 29T^{2}
37 1+0.874T+37T2 1 + 0.874T + 37T^{2}
41 1+1.23T+41T2 1 + 1.23T + 41T^{2}
43 1−9.69T+43T2 1 - 9.69T + 43T^{2}
47 1−1.70T+47T2 1 - 1.70T + 47T^{2}
53 1+11.1T+53T2 1 + 11.1T + 53T^{2}
59 1+3.70T+59T2 1 + 3.70T + 59T^{2}
61 1−13.1T+61T2 1 - 13.1T + 61T^{2}
67 1−0.763T+67T2 1 - 0.763T + 67T^{2}
71 1+8.94T+71T2 1 + 8.94T + 71T^{2}
73 1−10.2T+73T2 1 - 10.2T + 73T^{2}
79 1−10.2T+79T2 1 - 10.2T + 79T^{2}
83 1−2.95T+83T2 1 - 2.95T + 83T^{2}
89 1−3.57T+89T2 1 - 3.57T + 89T^{2}
97 1−15.4T+97T2 1 - 15.4T + 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

−7.896895953905557748334841924972, −7.30185439221329288255183511433, −6.51295086038913962638315405964, −5.88312682765922129578987255439, −4.77218630312413135903433983445, −3.84007104320514662068717778288, −3.32485067809093110023620106391, −2.63440649070297370809374326545, −1.78432389997571686180961675643, −0.72021458917826176058807563263, 0.72021458917826176058807563263, 1.78432389997571686180961675643, 2.63440649070297370809374326545, 3.32485067809093110023620106391, 3.84007104320514662068717778288, 4.77218630312413135903433983445, 5.88312682765922129578987255439, 6.51295086038913962638315405964, 7.30185439221329288255183511433, 7.896895953905557748334841924972

Graph of the ZZ-function along the critical line