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Properties

Label 2-312-1.1-c3-0-17
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 + 3.29·5-s − 25.8·7-s + 9·9-s − 8.83·11-s − 13·13-s + 9.87·15-s − 10.2·17-s − 119.·19-s − 77.6·21-s − 141.·23-s − 114.·25-s + 27·27-s + 170.·29-s + 226.·31-s − 26.5·33-s − 85.1·35-s − 225.·37-s − 39·39-s − 274.·41-s − 111.·43-s + 29.6·45-s + 156.·47-s + 326.·49-s − 30.7·51-s − 85.1·53-s − 29.0·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.294·5-s − 1.39·7-s + 0.333·9-s − 0.242·11-s − 0.277·13-s + 0.169·15-s − 0.146·17-s − 1.44·19-s − 0.806·21-s − 1.28·23-s − 0.913·25-s + 0.192·27-s + 1.09·29-s + 1.31·31-s − 0.139·33-s − 0.411·35-s − 1.00·37-s − 0.160·39-s − 1.04·41-s − 0.394·43-s + 0.0981·45-s + 0.485·47-s + 0.951·49-s − 0.0844·51-s − 0.220·53-s − 0.0712·55-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 1−3.29T+125T2 1 - 3.29T + 125T^{2}
7 1+25.8T+343T2 1 + 25.8T + 343T^{2}
11 1+8.83T+1.33e3T2 1 + 8.83T + 1.33e3T^{2}
17 1+10.2T+4.91e3T2 1 + 10.2T + 4.91e3T^{2}
19 1+119.T+6.85e3T2 1 + 119.T + 6.85e3T^{2}
23 1+141.T+1.21e4T2 1 + 141.T + 1.21e4T^{2}
29 1−170.T+2.43e4T2 1 - 170.T + 2.43e4T^{2}
31 1−226.T+2.97e4T2 1 - 226.T + 2.97e4T^{2}
37 1+225.T+5.06e4T2 1 + 225.T + 5.06e4T^{2}
41 1+274.T+6.89e4T2 1 + 274.T + 6.89e4T^{2}
43 1+111.T+7.95e4T2 1 + 111.T + 7.95e4T^{2}
47 1−156.T+1.03e5T2 1 - 156.T + 1.03e5T^{2}
53 1+85.1T+1.48e5T2 1 + 85.1T + 1.48e5T^{2}
59 1+889.T+2.05e5T2 1 + 889.T + 2.05e5T^{2}
61 1−463.T+2.26e5T2 1 - 463.T + 2.26e5T^{2}
67 1+459.T+3.00e5T2 1 + 459.T + 3.00e5T^{2}
71 1+560.T+3.57e5T2 1 + 560.T + 3.57e5T^{2}
73 1−784.T+3.89e5T2 1 - 784.T + 3.89e5T^{2}
79 1−241.T+4.93e5T2 1 - 241.T + 4.93e5T^{2}
83 1+1.27e3T+5.71e5T2 1 + 1.27e3T + 5.71e5T^{2}
89 1−1.08e3T+7.04e5T2 1 - 1.08e3T + 7.04e5T^{2}
97 1+79.9T+9.12e5T2 1 + 79.9T + 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

−10.32562671646902159028816148497, −9.993498419882439564928277611599, −8.911621503580280151647705856309, −8.028751454148506341239027958314, −6.73373998475477755736772247052, −6.05350067582724653057719832987, −4.45692626904635966156228758992, −3.26800874539872326447758293689, −2.13155032319669596782182198073, 0, 2.13155032319669596782182198073, 3.26800874539872326447758293689, 4.45692626904635966156228758992, 6.05350067582724653057719832987, 6.73373998475477755736772247052, 8.028751454148506341239027958314, 8.911621503580280151647705856309, 9.993498419882439564928277611599, 10.32562671646902159028816148497

Graph of the ZZ-function along the critical line