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Properties

Label 2-312-1.1-c3-0-7
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 + 3.18·5-s + 23.9·7-s + 9·9-s − 6.74·11-s + 13·13-s + 9.55·15-s + 104.·17-s − 137.·19-s + 71.8·21-s + 110.·23-s − 114.·25-s + 27·27-s − 57.6·29-s + 319.·31-s − 20.2·33-s + 76.2·35-s − 2.88·37-s + 39·39-s + 319.·41-s + 344.·43-s + 28.6·45-s − 439.·47-s + 229.·49-s + 312.·51-s − 97.2·53-s − 21.5·55-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.285·5-s + 1.29·7-s + 0.333·9-s − 0.184·11-s + 0.277·13-s + 0.164·15-s + 1.48·17-s − 1.66·19-s + 0.746·21-s + 0.999·23-s − 0.918·25-s + 0.192·27-s − 0.368·29-s + 1.85·31-s − 0.106·33-s + 0.368·35-s − 0.0128·37-s + 0.160·39-s + 1.21·41-s + 1.22·43-s + 0.0950·45-s − 1.36·47-s + 0.670·49-s + 0.858·51-s − 0.252·53-s − 0.0527·55-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 2.8217220392.821722039
L(12)L(\frac12) ≈\approx 2.8217220392.821722039
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 1−3.18T+125T2 1 - 3.18T + 125T^{2}
7 1−23.9T+343T2 1 - 23.9T + 343T^{2}
11 1+6.74T+1.33e3T2 1 + 6.74T + 1.33e3T^{2}
17 1−104.T+4.91e3T2 1 - 104.T + 4.91e3T^{2}
19 1+137.T+6.85e3T2 1 + 137.T + 6.85e3T^{2}
23 1−110.T+1.21e4T2 1 - 110.T + 1.21e4T^{2}
29 1+57.6T+2.43e4T2 1 + 57.6T + 2.43e4T^{2}
31 1−319.T+2.97e4T2 1 - 319.T + 2.97e4T^{2}
37 1+2.88T+5.06e4T2 1 + 2.88T + 5.06e4T^{2}
41 1−319.T+6.89e4T2 1 - 319.T + 6.89e4T^{2}
43 1−344.T+7.95e4T2 1 - 344.T + 7.95e4T^{2}
47 1+439.T+1.03e5T2 1 + 439.T + 1.03e5T^{2}
53 1+97.2T+1.48e5T2 1 + 97.2T + 1.48e5T^{2}
59 1−448.T+2.05e5T2 1 - 448.T + 2.05e5T^{2}
61 1+264.T+2.26e5T2 1 + 264.T + 2.26e5T^{2}
67 1+712.T+3.00e5T2 1 + 712.T + 3.00e5T^{2}
71 1−1.13e3T+3.57e5T2 1 - 1.13e3T + 3.57e5T^{2}
73 1+666.T+3.89e5T2 1 + 666.T + 3.89e5T^{2}
79 1−828.T+4.93e5T2 1 - 828.T + 4.93e5T^{2}
83 1−734.T+5.71e5T2 1 - 734.T + 5.71e5T^{2}
89 1−153.T+7.04e5T2 1 - 153.T + 7.04e5T^{2}
97 1+569.T+9.12e5T2 1 + 569.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

−11.15003699293305770209381085460, −10.34327466307908007684873644831, −9.298997621017830567667931692679, −8.231348745215258846556215206362, −7.74625267847375434307166331133, −6.32463786780079316901959010933, −5.13008107496469887309850665877, −4.05860914379598049431676422649, −2.55081009997226885584129771962, −1.29496283448469993975045199689, 1.29496283448469993975045199689, 2.55081009997226885584129771962, 4.05860914379598049431676422649, 5.13008107496469887309850665877, 6.32463786780079316901959010933, 7.74625267847375434307166331133, 8.231348745215258846556215206362, 9.298997621017830567667931692679, 10.34327466307908007684873644831, 11.15003699293305770209381085460

Graph of the ZZ-function along the critical line