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Properties

Label 2-1216-1.1-c3-0-42
Degree 22
Conductor 12161216
Sign 11
Analytic cond. 71.746371.7463
Root an. cond. 8.470328.47032
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-s + 8·5-s − 17·7-s − 26·9-s + 70·11-s + 61·13-s + 8·15-s + 83·17-s − 19·19-s − 17·21-s + 115·23-s − 61·25-s − 53·27-s − 279·29-s − 72·31-s + 70·33-s − 136·35-s + 34·37-s + 61·39-s + 108·41-s + 192·43-s − 208·45-s − 392·47-s − 54·49-s + 83·51-s − 131·53-s + 560·55-s + ⋯
L(s)  = 1  + 0.192·3-s + 0.715·5-s − 0.917·7-s − 0.962·9-s + 1.91·11-s + 1.30·13-s + 0.137·15-s + 1.18·17-s − 0.229·19-s − 0.176·21-s + 1.04·23-s − 0.487·25-s − 0.377·27-s − 1.78·29-s − 0.417·31-s + 0.369·33-s − 0.656·35-s + 0.151·37-s + 0.250·39-s + 0.411·41-s + 0.680·43-s − 0.689·45-s − 1.21·47-s − 0.157·49-s + 0.227·51-s − 0.339·53-s + 1.37·55-s + ⋯

Functional equation

Λ(s)=(1216s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1216s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 12161216    =    26⋅192^{6} \cdot 19
Sign: 11
Analytic conductor: 71.746371.7463
Root analytic conductor: 8.470328.47032
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1216, ( :3/2), 1)(2,\ 1216,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 2.6901729772.690172977
L(12)L(\frac12) ≈\approx 2.6901729772.690172977
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
19 1+pT 1 + p T
good3 1−T+p3T2 1 - T + p^{3} T^{2}
5 1−8T+p3T2 1 - 8 T + p^{3} T^{2}
7 1+17T+p3T2 1 + 17 T + p^{3} T^{2}
11 1−70T+p3T2 1 - 70 T + p^{3} T^{2}
13 1−61T+p3T2 1 - 61 T + p^{3} T^{2}
17 1−83T+p3T2 1 - 83 T + p^{3} T^{2}
23 1−5pT+p3T2 1 - 5 p T + p^{3} T^{2}
29 1+279T+p3T2 1 + 279 T + p^{3} T^{2}
31 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
37 1−34T+p3T2 1 - 34 T + p^{3} T^{2}
41 1−108T+p3T2 1 - 108 T + p^{3} T^{2}
43 1−192T+p3T2 1 - 192 T + p^{3} T^{2}
47 1+392T+p3T2 1 + 392 T + p^{3} T^{2}
53 1+131T+p3T2 1 + 131 T + p^{3} T^{2}
59 1−609T+p3T2 1 - 609 T + p^{3} T^{2}
61 1+338T+p3T2 1 + 338 T + p^{3} T^{2}
67 1−461T+p3T2 1 - 461 T + p^{3} T^{2}
71 1−750T+p3T2 1 - 750 T + p^{3} T^{2}
73 1−1177T+p3T2 1 - 1177 T + p^{3} T^{2}
79 1+22T+p3T2 1 + 22 T + p^{3} T^{2}
83 1−810T+p3T2 1 - 810 T + p^{3} T^{2}
89 1+476T+p3T2 1 + 476 T + p^{3} T^{2}
97 1−1426T+p3T2 1 - 1426 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

−9.276233092838817232407378248251, −8.864140576354861380061253434329, −7.76607992946098285409730474520, −6.56817419297560162083123762057, −6.14067386863665930609530442811, −5.37888289565725550576091196400, −3.71819781474324443672830356922, −3.41039964425079254379869351903, −1.94755849058392082225080345299, −0.861851611062581382759056021152, 0.861851611062581382759056021152, 1.94755849058392082225080345299, 3.41039964425079254379869351903, 3.71819781474324443672830356922, 5.37888289565725550576091196400, 6.14067386863665930609530442811, 6.56817419297560162083123762057, 7.76607992946098285409730474520, 8.864140576354861380061253434329, 9.276233092838817232407378248251

Graph of the ZZ-function along the critical line