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Properties

Label 2-396-1.1-c3-0-6
Degree 22
Conductor 396396
Sign 11
Analytic cond. 23.364723.3647
Root an. cond. 4.833714.83371
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  + 12·5-s + 26·7-s + 11·11-s − 34·13-s + 126·17-s + 110·19-s − 180·23-s + 19·25-s − 18·29-s − 292·31-s + 312·35-s − 238·37-s + 426·41-s + 146·43-s + 528·47-s + 333·49-s + 408·53-s + 132·55-s + 324·59-s − 550·61-s − 408·65-s + 824·67-s + 552·71-s − 850·73-s + 286·77-s + 866·79-s − 660·83-s + ⋯
L(s)  = 1  + 1.07·5-s + 1.40·7-s + 0.301·11-s − 0.725·13-s + 1.79·17-s + 1.32·19-s − 1.63·23-s + 0.151·25-s − 0.115·29-s − 1.69·31-s + 1.50·35-s − 1.05·37-s + 1.62·41-s + 0.517·43-s + 1.63·47-s + 0.970·49-s + 1.05·53-s + 0.323·55-s + 0.714·59-s − 1.15·61-s − 0.778·65-s + 1.50·67-s + 0.922·71-s − 1.36·73-s + 0.423·77-s + 1.23·79-s − 0.872·83-s + ⋯

Functional equation

Λ(s)=(396s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(396s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 396 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 396396    =    22⋅32⋅112^{2} \cdot 3^{2} \cdot 11
Sign: 11
Analytic conductor: 23.364723.3647
Root analytic conductor: 4.833714.83371
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 396, ( :3/2), 1)(2,\ 396,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.8269730342.826973034
L(12)L(\frac12) ≈\approx 2.8269730342.826973034
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 1
11 1−pT 1 - p T
good5 1−12T+p3T2 1 - 12 T + p^{3} T^{2}
7 1−26T+p3T2 1 - 26 T + p^{3} T^{2}
13 1+34T+p3T2 1 + 34 T + p^{3} T^{2}
17 1−126T+p3T2 1 - 126 T + p^{3} T^{2}
19 1−110T+p3T2 1 - 110 T + p^{3} T^{2}
23 1+180T+p3T2 1 + 180 T + p^{3} T^{2}
29 1+18T+p3T2 1 + 18 T + p^{3} T^{2}
31 1+292T+p3T2 1 + 292 T + p^{3} T^{2}
37 1+238T+p3T2 1 + 238 T + p^{3} T^{2}
41 1−426T+p3T2 1 - 426 T + p^{3} T^{2}
43 1−146T+p3T2 1 - 146 T + p^{3} T^{2}
47 1−528T+p3T2 1 - 528 T + p^{3} T^{2}
53 1−408T+p3T2 1 - 408 T + p^{3} T^{2}
59 1−324T+p3T2 1 - 324 T + p^{3} T^{2}
61 1+550T+p3T2 1 + 550 T + p^{3} T^{2}
67 1−824T+p3T2 1 - 824 T + p^{3} T^{2}
71 1−552T+p3T2 1 - 552 T + p^{3} T^{2}
73 1+850T+p3T2 1 + 850 T + p^{3} T^{2}
79 1−866T+p3T2 1 - 866 T + p^{3} T^{2}
83 1+660T+p3T2 1 + 660 T + p^{3} T^{2}
89 1−768T+p3T2 1 - 768 T + p^{3} T^{2}
97 1+286T+p3T2 1 + 286 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

−10.78088469528999660045500502690, −9.900247032872803534339008285340, −9.228631475715792278703973366914, −7.930120460337533924489539798647, −7.34181109182134643110366964987, −5.66307497167811783470324826673, −5.37736989611351380281319731735, −3.89401589279563103810546676562, −2.26441803689400852130289245883, −1.25386749019618991502257910908, 1.25386749019618991502257910908, 2.26441803689400852130289245883, 3.89401589279563103810546676562, 5.37736989611351380281319731735, 5.66307497167811783470324826673, 7.34181109182134643110366964987, 7.930120460337533924489539798647, 9.228631475715792278703973366914, 9.900247032872803534339008285340, 10.78088469528999660045500502690

Graph of the ZZ-function along the critical line