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Properties

Label 2-2496-1.1-c3-0-111
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 − 4·5-s − 4·7-s + 9·9-s + 2·11-s + 13·13-s − 12·15-s − 6·17-s − 36·19-s − 12·21-s + 20·23-s − 109·25-s + 27·27-s + 14·29-s + 152·31-s + 6·33-s + 16·35-s + 258·37-s + 39·39-s + 84·41-s − 188·43-s − 36·45-s − 254·47-s − 327·49-s − 18·51-s − 366·53-s − 8·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.357·5-s − 0.215·7-s + 1/3·9-s + 0.0548·11-s + 0.277·13-s − 0.206·15-s − 0.0856·17-s − 0.434·19-s − 0.124·21-s + 0.181·23-s − 0.871·25-s + 0.192·27-s + 0.0896·29-s + 0.880·31-s + 0.0316·33-s + 0.0772·35-s + 1.14·37-s + 0.160·39-s + 0.319·41-s − 0.666·43-s − 0.119·45-s − 0.788·47-s − 0.953·49-s − 0.0494·51-s − 0.948·53-s − 0.0196·55-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+4T+p3T2 1 + 4 T + p^{3} T^{2}
7 1+4T+p3T2 1 + 4 T + p^{3} T^{2}
11 1−2T+p3T2 1 - 2 T + p^{3} T^{2}
17 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
19 1+36T+p3T2 1 + 36 T + p^{3} T^{2}
23 1−20T+p3T2 1 - 20 T + p^{3} T^{2}
29 1−14T+p3T2 1 - 14 T + p^{3} T^{2}
31 1−152T+p3T2 1 - 152 T + p^{3} T^{2}
37 1−258T+p3T2 1 - 258 T + p^{3} T^{2}
41 1−84T+p3T2 1 - 84 T + p^{3} T^{2}
43 1+188T+p3T2 1 + 188 T + p^{3} T^{2}
47 1+254T+p3T2 1 + 254 T + p^{3} T^{2}
53 1+366T+p3T2 1 + 366 T + p^{3} T^{2}
59 1−550T+p3T2 1 - 550 T + p^{3} T^{2}
61 1−14T+p3T2 1 - 14 T + p^{3} T^{2}
67 1−448T+p3T2 1 - 448 T + p^{3} T^{2}
71 1+926T+p3T2 1 + 926 T + p^{3} T^{2}
73 1−254T+p3T2 1 - 254 T + p^{3} T^{2}
79 1+1328T+p3T2 1 + 1328 T + p^{3} T^{2}
83 1−186T+p3T2 1 - 186 T + p^{3} T^{2}
89 1+336T+p3T2 1 + 336 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

−8.199007141859573397779797155115, −7.58885355056681573196717188067, −6.65990914435576323876706701076, −6.01069391225621990630018905738, −4.85877890101840951485711166306, −4.09208728516349750294318093559, −3.27927482062737414944372447471, −2.38445835137681607404900500521, −1.27838234338954444173473271919, 0, 1.27838234338954444173473271919, 2.38445835137681607404900500521, 3.27927482062737414944372447471, 4.09208728516349750294318093559, 4.85877890101840951485711166306, 6.01069391225621990630018905738, 6.65990914435576323876706701076, 7.58885355056681573196717188067, 8.199007141859573397779797155115

Graph of the ZZ-function along the critical line