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Properties

Label 2-31e2-1.1-c1-0-21
Degree 22
Conductor 961961
Sign 11
Analytic cond. 7.673627.67362
Root an. cond. 2.770132.77013
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  + 2.49·2-s − 2.47·3-s + 4.23·4-s − 1.14·5-s − 6.17·6-s + 0.265·7-s + 5.56·8-s + 3.11·9-s − 2.86·10-s − 0.291·11-s − 10.4·12-s + 4.52·13-s + 0.661·14-s + 2.84·15-s + 5.44·16-s + 5.49·17-s + 7.77·18-s + 6.95·19-s − 4.86·20-s − 0.655·21-s − 0.727·22-s + 2.96·23-s − 13.7·24-s − 3.67·25-s + 11.2·26-s − 0.286·27-s + 1.12·28-s + ⋯
L(s)  = 1  + 1.76·2-s − 1.42·3-s + 2.11·4-s − 0.513·5-s − 2.52·6-s + 0.100·7-s + 1.96·8-s + 1.03·9-s − 0.907·10-s − 0.0879·11-s − 3.02·12-s + 1.25·13-s + 0.176·14-s + 0.733·15-s + 1.36·16-s + 1.33·17-s + 1.83·18-s + 1.59·19-s − 1.08·20-s − 0.143·21-s − 0.155·22-s + 0.618·23-s − 2.81·24-s − 0.735·25-s + 2.21·26-s − 0.0552·27-s + 0.211·28-s + ⋯

Functional equation

Λ(s)=(961s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(961s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 961 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 961961    =    31231^{2}
Sign: 11
Analytic conductor: 7.673627.67362
Root analytic conductor: 2.770132.77013
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 961, ( :1/2), 1)(2,\ 961,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.8814405652.881440565
L(12)L(\frac12) ≈\approx 2.8814405652.881440565
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)
bad31 1 1
good2 1−2.49T+2T2 1 - 2.49T + 2T^{2}
3 1+2.47T+3T2 1 + 2.47T + 3T^{2}
5 1+1.14T+5T2 1 + 1.14T + 5T^{2}
7 1−0.265T+7T2 1 - 0.265T + 7T^{2}
11 1+0.291T+11T2 1 + 0.291T + 11T^{2}
13 1−4.52T+13T2 1 - 4.52T + 13T^{2}
17 1−5.49T+17T2 1 - 5.49T + 17T^{2}
19 1−6.95T+19T2 1 - 6.95T + 19T^{2}
23 1−2.96T+23T2 1 - 2.96T + 23T^{2}
29 1+3.32T+29T2 1 + 3.32T + 29T^{2}
37 1+4.07T+37T2 1 + 4.07T + 37T^{2}
41 1−7.49T+41T2 1 - 7.49T + 41T^{2}
43 1−8.05T+43T2 1 - 8.05T + 43T^{2}
47 1−5.51T+47T2 1 - 5.51T + 47T^{2}
53 1+4.75T+53T2 1 + 4.75T + 53T^{2}
59 1+2.35T+59T2 1 + 2.35T + 59T^{2}
61 1+10.0T+61T2 1 + 10.0T + 61T^{2}
67 1+3.58T+67T2 1 + 3.58T + 67T^{2}
71 1−10.8T+71T2 1 - 10.8T + 71T^{2}
73 1+2.27T+73T2 1 + 2.27T + 73T^{2}
79 1+3.79T+79T2 1 + 3.79T + 79T^{2}
83 1−2.79T+83T2 1 - 2.79T + 83T^{2}
89 1+3.20T+89T2 1 + 3.20T + 89T^{2}
97 1−2.01T+97T2 1 - 2.01T + 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

−10.71266341049430674537478092850, −9.428833947251081132681148322140, −7.83244653316419576330617568411, −7.14490477213935664124002624052, −6.04396100125625085608745799301, −5.67179948428029802495443929114, −4.91735484067636694032458029244, −3.91267720446921018587512311540, −3.13013962216912147376389114347, −1.22757920293218799713496602599, 1.22757920293218799713496602599, 3.13013962216912147376389114347, 3.91267720446921018587512311540, 4.91735484067636694032458029244, 5.67179948428029802495443929114, 6.04396100125625085608745799301, 7.14490477213935664124002624052, 7.83244653316419576330617568411, 9.428833947251081132681148322140, 10.71266341049430674537478092850

Graph of the ZZ-function along the critical line