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Properties

Label 2-2496-1.1-c3-0-118
Degree 22
Conductor 24962496
Sign −1-1
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 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 6·5-s + 20·7-s + 9·9-s − 24·11-s − 13·13-s − 18·15-s − 30·17-s + 16·19-s + 60·21-s − 72·23-s − 89·25-s + 27·27-s + 282·29-s + 164·31-s − 72·33-s − 120·35-s − 110·37-s − 39·39-s − 126·41-s − 164·43-s − 54·45-s − 204·47-s + 57·49-s − 90·51-s + 738·53-s + 144·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.536·5-s + 1.07·7-s + 1/3·9-s − 0.657·11-s − 0.277·13-s − 0.309·15-s − 0.428·17-s + 0.193·19-s + 0.623·21-s − 0.652·23-s − 0.711·25-s + 0.192·27-s + 1.80·29-s + 0.950·31-s − 0.379·33-s − 0.579·35-s − 0.488·37-s − 0.160·39-s − 0.479·41-s − 0.581·43-s − 0.178·45-s − 0.633·47-s + 0.166·49-s − 0.247·51-s + 1.91·53-s + 0.353·55-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: −1-1
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: 11
Selberg data: (2, 2496, ( :3/2), −1)(2,\ 2496,\ (\ :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−pT 1 - p T
13 1+pT 1 + p T
good5 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
7 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
11 1+24T+p3T2 1 + 24 T + p^{3} T^{2}
17 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
19 1−16T+p3T2 1 - 16 T + p^{3} T^{2}
23 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
29 1−282T+p3T2 1 - 282 T + p^{3} T^{2}
31 1−164T+p3T2 1 - 164 T + p^{3} T^{2}
37 1+110T+p3T2 1 + 110 T + p^{3} T^{2}
41 1+126T+p3T2 1 + 126 T + p^{3} T^{2}
43 1+164T+p3T2 1 + 164 T + p^{3} T^{2}
47 1+204T+p3T2 1 + 204 T + p^{3} T^{2}
53 1−738T+p3T2 1 - 738 T + p^{3} T^{2}
59 1+120T+p3T2 1 + 120 T + p^{3} T^{2}
61 1+614T+p3T2 1 + 614 T + p^{3} T^{2}
67 1+848T+p3T2 1 + 848 T + p^{3} T^{2}
71 1−132T+p3T2 1 - 132 T + p^{3} T^{2}
73 1−218T+p3T2 1 - 218 T + p^{3} T^{2}
79 1+1096T+p3T2 1 + 1096 T + p^{3} T^{2}
83 1+552T+p3T2 1 + 552 T + p^{3} T^{2}
89 1−210T+p3T2 1 - 210 T + p^{3} T^{2}
97 1+1726T+p3T2 1 + 1726 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

−8.332000882195298676902530724457, −7.60183396981261854466321704808, −6.86963171070988769960099753345, −5.79212091346365461341625292036, −4.77004707883429040554838275059, −4.33308202003223622669385089414, −3.19130007233440145044251208007, −2.32823925710184006128101027239, −1.34199051385831543847779805331, 0, 1.34199051385831543847779805331, 2.32823925710184006128101027239, 3.19130007233440145044251208007, 4.33308202003223622669385089414, 4.77004707883429040554838275059, 5.79212091346365461341625292036, 6.86963171070988769960099753345, 7.60183396981261854466321704808, 8.332000882195298676902530724457

Graph of the ZZ-function along the critical line