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Properties

Label 2-312-1.1-c3-0-8
Degree 22
Conductor 312312
Sign 11
Analytic cond. 18.408518.4085
Root an. cond. 4.290524.29052
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  + 3·3-s + 12.9·5-s + 1.76·7-s + 9·9-s + 25.1·11-s + 13·13-s + 38.8·15-s − 62.6·17-s + 139.·19-s + 5.29·21-s − 56.6·23-s + 42.9·25-s + 27·27-s + 75.4·29-s − 71.2·31-s + 75.5·33-s + 22.8·35-s + 55.7·37-s + 39·39-s − 40.0·41-s + 14.7·43-s + 116.·45-s + 531.·47-s − 339.·49-s − 187.·51-s + 368.·53-s + 326.·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.15·5-s + 0.0952·7-s + 0.333·9-s + 0.690·11-s + 0.277·13-s + 0.669·15-s − 0.893·17-s + 1.68·19-s + 0.0550·21-s − 0.513·23-s + 0.343·25-s + 0.192·27-s + 0.482·29-s − 0.413·31-s + 0.398·33-s + 0.110·35-s + 0.247·37-s + 0.160·39-s − 0.152·41-s + 0.0524·43-s + 0.386·45-s + 1.64·47-s − 0.990·49-s − 0.515·51-s + 0.953·53-s + 0.800·55-s + ⋯

Functional equation

Λ(s)=(312s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(312s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 312312    =    23⋅3⋅132^{3} \cdot 3 \cdot 13
Sign: 11
Analytic conductor: 18.408518.4085
Root analytic conductor: 4.290524.29052
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 312, ( :3/2), 1)(2,\ 312,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.9787666822.978766682
L(12)L(\frac12) ≈\approx 2.9787666822.978766682
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
3 1−3T 1 - 3T
13 1−13T 1 - 13T
good5 1−12.9T+125T2 1 - 12.9T + 125T^{2}
7 1−1.76T+343T2 1 - 1.76T + 343T^{2}
11 1−25.1T+1.33e3T2 1 - 25.1T + 1.33e3T^{2}
17 1+62.6T+4.91e3T2 1 + 62.6T + 4.91e3T^{2}
19 1−139.T+6.85e3T2 1 - 139.T + 6.85e3T^{2}
23 1+56.6T+1.21e4T2 1 + 56.6T + 1.21e4T^{2}
29 1−75.4T+2.43e4T2 1 - 75.4T + 2.43e4T^{2}
31 1+71.2T+2.97e4T2 1 + 71.2T + 2.97e4T^{2}
37 1−55.7T+5.06e4T2 1 - 55.7T + 5.06e4T^{2}
41 1+40.0T+6.89e4T2 1 + 40.0T + 6.89e4T^{2}
43 1−14.7T+7.95e4T2 1 - 14.7T + 7.95e4T^{2}
47 1−531.T+1.03e5T2 1 - 531.T + 1.03e5T^{2}
53 1−368.T+1.48e5T2 1 - 368.T + 1.48e5T^{2}
59 1−165.T+2.05e5T2 1 - 165.T + 2.05e5T^{2}
61 1+145.T+2.26e5T2 1 + 145.T + 2.26e5T^{2}
67 1−901.T+3.00e5T2 1 - 901.T + 3.00e5T^{2}
71 1+345.T+3.57e5T2 1 + 345.T + 3.57e5T^{2}
73 1+292.T+3.89e5T2 1 + 292.T + 3.89e5T^{2}
79 1+722.T+4.93e5T2 1 + 722.T + 4.93e5T^{2}
83 1−565.T+5.71e5T2 1 - 565.T + 5.71e5T^{2}
89 1+275.T+7.04e5T2 1 + 275.T + 7.04e5T^{2}
97 1+1.82e3T+9.12e5T2 1 + 1.82e3T + 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

−11.20773849669217654583099184191, −10.04958611653186648911677439601, −9.405473028016121317522936551687, −8.606289509820526759436802645793, −7.36384634085107473617108983116, −6.33426764879916427070692880253, −5.31523515560876524689158029908, −3.92940301469109970451363992625, −2.54734859652549177684038742721, −1.34536899869049717706855553257, 1.34536899869049717706855553257, 2.54734859652549177684038742721, 3.92940301469109970451363992625, 5.31523515560876524689158029908, 6.33426764879916427070692880253, 7.36384634085107473617108983116, 8.606289509820526759436802645793, 9.405473028016121317522936551687, 10.04958611653186648911677439601, 11.20773849669217654583099184191

Graph of the ZZ-function along the critical line