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Properties

Label 2-312-1.1-c3-0-14
Degree 22
Conductor 312312
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 13.1·5-s + 15.6·7-s + 9·9-s − 55.7·11-s + 13·13-s − 39.3·15-s + 23.4·17-s − 25.6·19-s + 46.8·21-s − 189.·23-s + 47.2·25-s + 27·27-s − 236.·29-s + 47.0·31-s − 167.·33-s − 204.·35-s − 154.·37-s + 39·39-s − 34.6·41-s − 398.·43-s − 118.·45-s + 582.·47-s − 99.1·49-s + 70.4·51-s − 361.·53-s + 731.·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.17·5-s + 0.843·7-s + 0.333·9-s − 1.52·11-s + 0.277·13-s − 0.677·15-s + 0.334·17-s − 0.309·19-s + 0.486·21-s − 1.71·23-s + 0.377·25-s + 0.192·27-s − 1.51·29-s + 0.272·31-s − 0.881·33-s − 0.989·35-s − 0.687·37-s + 0.160·39-s − 0.132·41-s − 1.41·43-s − 0.391·45-s + 1.80·47-s − 0.289·49-s + 0.193·51-s − 0.935·53-s + 1.79·55-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: −1-1
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: 11
Selberg data: (2, 312, ( :3/2), −1)(2,\ 312,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
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+13.1T+125T2 1 + 13.1T + 125T^{2}
7 1−15.6T+343T2 1 - 15.6T + 343T^{2}
11 1+55.7T+1.33e3T2 1 + 55.7T + 1.33e3T^{2}
17 1−23.4T+4.91e3T2 1 - 23.4T + 4.91e3T^{2}
19 1+25.6T+6.85e3T2 1 + 25.6T + 6.85e3T^{2}
23 1+189.T+1.21e4T2 1 + 189.T + 1.21e4T^{2}
29 1+236.T+2.43e4T2 1 + 236.T + 2.43e4T^{2}
31 1−47.0T+2.97e4T2 1 - 47.0T + 2.97e4T^{2}
37 1+154.T+5.06e4T2 1 + 154.T + 5.06e4T^{2}
41 1+34.6T+6.89e4T2 1 + 34.6T + 6.89e4T^{2}
43 1+398.T+7.95e4T2 1 + 398.T + 7.95e4T^{2}
47 1−582.T+1.03e5T2 1 - 582.T + 1.03e5T^{2}
53 1+361.T+1.48e5T2 1 + 361.T + 1.48e5T^{2}
59 1−396.T+2.05e5T2 1 - 396.T + 2.05e5T^{2}
61 1+211.T+2.26e5T2 1 + 211.T + 2.26e5T^{2}
67 1−85.8T+3.00e5T2 1 - 85.8T + 3.00e5T^{2}
71 1+651.T+3.57e5T2 1 + 651.T + 3.57e5T^{2}
73 1−927.T+3.89e5T2 1 - 927.T + 3.89e5T^{2}
79 1+1.11e3T+4.93e5T2 1 + 1.11e3T + 4.93e5T^{2}
83 1−391.T+5.71e5T2 1 - 391.T + 5.71e5T^{2}
89 1+745.T+7.04e5T2 1 + 745.T + 7.04e5T^{2}
97 1+173.T+9.12e5T2 1 + 173.T + 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

−10.83394782654989645206140862298, −9.930619319756160097953807829626, −8.486485898348483876344336995153, −7.985894070032192047860297427511, −7.32760487574813187890454752491, −5.63734182546104978125567275658, −4.46759038480210206541089976156, −3.45709916490424119660510429234, −2.00838632588726881683853527300, 0, 2.00838632588726881683853527300, 3.45709916490424119660510429234, 4.46759038480210206541089976156, 5.63734182546104978125567275658, 7.32760487574813187890454752491, 7.985894070032192047860297427511, 8.486485898348483876344336995153, 9.930619319756160097953807829626, 10.83394782654989645206140862298

Graph of the ZZ-function along the critical line