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Properties

Label 2-308-1.1-c3-0-6
Degree 22
Conductor 308308
Sign 11
Analytic cond. 18.172518.1725
Root an. cond. 4.262934.26293
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.16·3-s − 10.7·5-s + 7·7-s + 57.0·9-s − 11·11-s + 48.7·13-s − 98.2·15-s + 87.5·17-s − 35.4·19-s + 64.1·21-s + 215.·23-s − 10.1·25-s + 275.·27-s − 0.722·29-s + 270.·31-s − 100.·33-s − 75.0·35-s − 27.3·37-s + 447.·39-s − 381.·41-s − 137.·43-s − 611.·45-s − 202.·47-s + 49·49-s + 802.·51-s − 443.·53-s + 117.·55-s + ⋯
L(s)  = 1  + 1.76·3-s − 0.958·5-s + 0.377·7-s + 2.11·9-s − 0.301·11-s + 1.04·13-s − 1.69·15-s + 1.24·17-s − 0.428·19-s + 0.666·21-s + 1.95·23-s − 0.0809·25-s + 1.96·27-s − 0.00462·29-s + 1.56·31-s − 0.532·33-s − 0.362·35-s − 0.121·37-s + 1.83·39-s − 1.45·41-s − 0.487·43-s − 2.02·45-s − 0.628·47-s + 0.142·49-s + 2.20·51-s − 1.14·53-s + 0.289·55-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: 11
Analytic conductor: 18.172518.1725
Root analytic conductor: 4.262934.26293
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 308, ( :3/2), 1)(2,\ 308,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 3.2615864103.261586410
L(12)L(\frac12) ≈\approx 3.2615864103.261586410
L(52)L(\frac{5}{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 1
7 1−7T 1 - 7T
11 1+11T 1 + 11T
good3 1−9.16T+27T2 1 - 9.16T + 27T^{2}
5 1+10.7T+125T2 1 + 10.7T + 125T^{2}
13 1−48.7T+2.19e3T2 1 - 48.7T + 2.19e3T^{2}
17 1−87.5T+4.91e3T2 1 - 87.5T + 4.91e3T^{2}
19 1+35.4T+6.85e3T2 1 + 35.4T + 6.85e3T^{2}
23 1−215.T+1.21e4T2 1 - 215.T + 1.21e4T^{2}
29 1+0.722T+2.43e4T2 1 + 0.722T + 2.43e4T^{2}
31 1−270.T+2.97e4T2 1 - 270.T + 2.97e4T^{2}
37 1+27.3T+5.06e4T2 1 + 27.3T + 5.06e4T^{2}
41 1+381.T+6.89e4T2 1 + 381.T + 6.89e4T^{2}
43 1+137.T+7.95e4T2 1 + 137.T + 7.95e4T^{2}
47 1+202.T+1.03e5T2 1 + 202.T + 1.03e5T^{2}
53 1+443.T+1.48e5T2 1 + 443.T + 1.48e5T^{2}
59 1+845.T+2.05e5T2 1 + 845.T + 2.05e5T^{2}
61 1−539.T+2.26e5T2 1 - 539.T + 2.26e5T^{2}
67 1+162.T+3.00e5T2 1 + 162.T + 3.00e5T^{2}
71 1−550.T+3.57e5T2 1 - 550.T + 3.57e5T^{2}
73 1−52.4T+3.89e5T2 1 - 52.4T + 3.89e5T^{2}
79 1−379.T+4.93e5T2 1 - 379.T + 4.93e5T^{2}
83 1−47.7T+5.71e5T2 1 - 47.7T + 5.71e5T^{2}
89 1−308.T+7.04e5T2 1 - 308.T + 7.04e5T^{2}
97 1−93.5T+9.12e5T2 1 - 93.5T + 9.12e5T^{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

−11.22933308702922281383284913452, −10.18412326019096677684076639157, −9.117083252204804305110323292738, −8.217702324866974658776318812029, −7.893132894470447775775084838663, −6.74022801205435713217253405932, −4.86432200548533070772898966671, −3.67724627246997365226688157246, −2.96899550574724924062191494725, −1.33881361221283851389659702196, 1.33881361221283851389659702196, 2.96899550574724924062191494725, 3.67724627246997365226688157246, 4.86432200548533070772898966671, 6.74022801205435713217253405932, 7.893132894470447775775084838663, 8.217702324866974658776318812029, 9.117083252204804305110323292738, 10.18412326019096677684076639157, 11.22933308702922281383284913452

Graph of the ZZ-function along the critical line