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Properties

Label 2-9610-1.1-c1-0-62
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 − 0.593·3-s + 4-s + 5-s + 0.593·6-s + 4.51·7-s − 8-s − 2.64·9-s − 10-s + 1.40·11-s − 0.593·12-s − 6.32·13-s − 4.51·14-s − 0.593·15-s + 16-s + 2.05·17-s + 2.64·18-s − 4.46·19-s + 20-s − 2.67·21-s − 1.40·22-s + 5.51·23-s + 0.593·24-s + 25-s + 6.32·26-s + 3.35·27-s + 4.51·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.342·3-s + 0.5·4-s + 0.447·5-s + 0.242·6-s + 1.70·7-s − 0.353·8-s − 0.882·9-s − 0.316·10-s + 0.424·11-s − 0.171·12-s − 1.75·13-s − 1.20·14-s − 0.153·15-s + 0.250·16-s + 0.498·17-s + 0.624·18-s − 1.02·19-s + 0.223·20-s − 0.584·21-s − 0.299·22-s + 1.14·23-s + 0.121·24-s + 0.200·25-s + 1.24·26-s + 0.645·27-s + 0.853·28-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 1.3977844431.397784443
L(12)L(\frac12) ≈\approx 1.3977844431.397784443
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+0.593T+3T2 1 + 0.593T + 3T^{2}
7 1−4.51T+7T2 1 - 4.51T + 7T^{2}
11 1−1.40T+11T2 1 - 1.40T + 11T^{2}
13 1+6.32T+13T2 1 + 6.32T + 13T^{2}
17 1−2.05T+17T2 1 - 2.05T + 17T^{2}
19 1+4.46T+19T2 1 + 4.46T + 19T^{2}
23 1−5.51T+23T2 1 - 5.51T + 23T^{2}
29 1+1.32T+29T2 1 + 1.32T + 29T^{2}
37 1−4.32T+37T2 1 - 4.32T + 37T^{2}
41 1+12.1T+41T2 1 + 12.1T + 41T^{2}
43 1−2.46T+43T2 1 - 2.46T + 43T^{2}
47 1+4.38T+47T2 1 + 4.38T + 47T^{2}
53 1−4.86T+53T2 1 - 4.86T + 53T^{2}
59 1+13.1T+59T2 1 + 13.1T + 59T^{2}
61 1−0.891T+61T2 1 - 0.891T + 61T^{2}
67 1+1.37T+67T2 1 + 1.37T + 67T^{2}
71 1−6.43T+71T2 1 - 6.43T + 71T^{2}
73 1−6.38T+73T2 1 - 6.38T + 73T^{2}
79 1−6.32T+79T2 1 - 6.32T + 79T^{2}
83 1−14.3T+83T2 1 - 14.3T + 83T^{2}
89 1−2.72T+89T2 1 - 2.72T + 89T^{2}
97 1+5.32T+97T2 1 + 5.32T + 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

−7.88376535277291617629802008542, −7.06342225885206576687340222261, −6.46451923297649950990820286324, −5.52160094273289584505844810303, −5.04447147858500260288430190317, −4.49693896397925506023444034312, −3.18395918701216956804597845260, −2.30169293783595763353235163495, −1.72603852848213864734430902772, −0.63823361404291572569200802238, 0.63823361404291572569200802238, 1.72603852848213864734430902772, 2.30169293783595763353235163495, 3.18395918701216956804597845260, 4.49693896397925506023444034312, 5.04447147858500260288430190317, 5.52160094273289584505844810303, 6.46451923297649950990820286324, 7.06342225885206576687340222261, 7.88376535277291617629802008542

Graph of the ZZ-function along the critical line