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Properties

Label 2-2496-1.1-c3-0-8
Degree 22
Conductor 24962496
Sign 11
Analytic cond. 147.268147.268
Root an. cond. 12.135412.1354
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 10·5-s − 8·7-s + 9·9-s − 40·11-s − 13·13-s + 30·15-s + 130·17-s + 20·19-s + 24·21-s − 25·25-s − 27·27-s + 18·29-s − 184·31-s + 120·33-s + 80·35-s + 74·37-s + 39·39-s − 362·41-s − 76·43-s − 90·45-s − 452·47-s − 279·49-s − 390·51-s − 382·53-s + 400·55-s − 60·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.894·5-s − 0.431·7-s + 1/3·9-s − 1.09·11-s − 0.277·13-s + 0.516·15-s + 1.85·17-s + 0.241·19-s + 0.249·21-s − 1/5·25-s − 0.192·27-s + 0.115·29-s − 1.06·31-s + 0.633·33-s + 0.386·35-s + 0.328·37-s + 0.160·39-s − 1.37·41-s − 0.269·43-s − 0.298·45-s − 1.40·47-s − 0.813·49-s − 1.07·51-s − 0.990·53-s + 0.980·55-s − 0.139·57-s + ⋯

Functional equation

Λ(s)=(2496s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(2496s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2496 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 24962496    =    26⋅3⋅132^{6} \cdot 3 \cdot 13
Sign: 11
Analytic conductor: 147.268147.268
Root analytic conductor: 12.135412.1354
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 2496, ( :3/2), 1)(2,\ 2496,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 0.53699215350.5369921535
L(12)L(\frac12) ≈\approx 0.53699215350.5369921535
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+pT 1 + p T
13 1+pT 1 + p T
good5 1+2pT+p3T2 1 + 2 p T + p^{3} T^{2}
7 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
11 1+40T+p3T2 1 + 40 T + p^{3} T^{2}
17 1−130T+p3T2 1 - 130 T + p^{3} T^{2}
19 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
23 1+p3T2 1 + p^{3} T^{2}
29 1−18T+p3T2 1 - 18 T + p^{3} T^{2}
31 1+184T+p3T2 1 + 184 T + p^{3} T^{2}
37 1−2pT+p3T2 1 - 2 p T + p^{3} T^{2}
41 1+362T+p3T2 1 + 362 T + p^{3} T^{2}
43 1+76T+p3T2 1 + 76 T + p^{3} T^{2}
47 1+452T+p3T2 1 + 452 T + p^{3} T^{2}
53 1+382T+p3T2 1 + 382 T + p^{3} T^{2}
59 1+464T+p3T2 1 + 464 T + p^{3} T^{2}
61 1+358T+p3T2 1 + 358 T + p^{3} T^{2}
67 1−700T+p3T2 1 - 700 T + p^{3} T^{2}
71 1+748T+p3T2 1 + 748 T + p^{3} T^{2}
73 1−1058T+p3T2 1 - 1058 T + p^{3} T^{2}
79 1+976T+p3T2 1 + 976 T + p^{3} T^{2}
83 1−1008T+p3T2 1 - 1008 T + p^{3} T^{2}
89 1+386T+p3T2 1 + 386 T + p^{3} T^{2}
97 1+614T+p3T2 1 + 614 T + p^{3} T^{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

−8.277128111182670127197665698516, −7.79470786174600877214805138615, −7.17912818826609614255064823452, −6.20215113234700980468648256545, −5.36678659725402263954052212756, −4.78544687286091458554004896539, −3.59468348153758132672877801260, −3.06410583350988850963273638139, −1.60893225071301858449973478687, −0.33245213530920080717427301061, 0.33245213530920080717427301061, 1.60893225071301858449973478687, 3.06410583350988850963273638139, 3.59468348153758132672877801260, 4.78544687286091458554004896539, 5.36678659725402263954052212756, 6.20215113234700980468648256545, 7.17912818826609614255064823452, 7.79470786174600877214805138615, 8.277128111182670127197665698516

Graph of the ZZ-function along the critical line