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Properties

Label 2-226512-1.1-c1-0-100
Degree 22
Conductor 226512226512
Sign −1-1
Analytic cond. 1808.701808.70
Root an. cond. 42.528942.5289
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s + 13-s − 3·17-s − 6·19-s + 7·23-s + 11·25-s − 9·29-s + 8·31-s + 2·37-s + 8·41-s + 7·43-s + 2·47-s − 7·49-s + 9·53-s + 4·59-s + 7·61-s − 4·65-s − 8·67-s − 14·71-s + 14·73-s + 5·79-s − 2·83-s + 12·85-s + 24·95-s − 4·97-s + 101-s + 103-s + ⋯
L(s)  = 1  − 1.78·5-s + 0.277·13-s − 0.727·17-s − 1.37·19-s + 1.45·23-s + 11/5·25-s − 1.67·29-s + 1.43·31-s + 0.328·37-s + 1.24·41-s + 1.06·43-s + 0.291·47-s − 49-s + 1.23·53-s + 0.520·59-s + 0.896·61-s − 0.496·65-s − 0.977·67-s − 1.66·71-s + 1.63·73-s + 0.562·79-s − 0.219·83-s + 1.30·85-s + 2.46·95-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + ⋯

Functional equation

Λ(s)=(226512s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(226512s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 226512226512    =    24⋅32⋅112⋅132^{4} \cdot 3^{2} \cdot 11^{2} \cdot 13
Sign: −1-1
Analytic conductor: 1808.701808.70
Root analytic conductor: 42.528942.5289
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 226512, ( :1/2), −1)(2,\ 226512,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
L(32)L(\frac{3}{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 1
13 1−T 1 - T
good5 1+4T+pT2 1 + 4 T + p T^{2}
7 1+pT2 1 + p T^{2}
17 1+3T+pT2 1 + 3 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1−7T+pT2 1 - 7 T + p T^{2}
29 1+9T+pT2 1 + 9 T + p T^{2}
31 1−8T+pT2 1 - 8 T + p T^{2}
37 1−2T+pT2 1 - 2 T + p T^{2}
41 1−8T+pT2 1 - 8 T + p T^{2}
43 1−7T+pT2 1 - 7 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1−9T+pT2 1 - 9 T + p T^{2}
59 1−4T+pT2 1 - 4 T + p T^{2}
61 1−7T+pT2 1 - 7 T + p T^{2}
67 1+8T+pT2 1 + 8 T + p T^{2}
71 1+14T+pT2 1 + 14 T + p T^{2}
73 1−14T+pT2 1 - 14 T + p T^{2}
79 1−5T+pT2 1 - 5 T + p T^{2}
83 1+2T+pT2 1 + 2 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1+4T+pT2 1 + 4 T + p 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

−13.08560551291501, −12.63655605428560, −12.34378622474458, −11.62872864165087, −11.26125944260169, −11.06118924484871, −10.58268413820921, −10.01068428406880, −9.095342933972631, −9.018082399431156, −8.423410613631972, −7.991034755490971, −7.512398903424604, −7.069806775527618, −6.626899416015057, −6.063632297236967, −5.417608524561375, −4.656706438831828, −4.417568563494208, −3.873558724142894, −3.496861810644455, −2.666437590578066, −2.354920643788847, −1.278321620359692, −0.6728150209483487, 0, 0.6728150209483487, 1.278321620359692, 2.354920643788847, 2.666437590578066, 3.496861810644455, 3.873558724142894, 4.417568563494208, 4.656706438831828, 5.417608524561375, 6.063632297236967, 6.626899416015057, 7.069806775527618, 7.512398903424604, 7.991034755490971, 8.423410613631972, 9.018082399431156, 9.095342933972631, 10.01068428406880, 10.58268413820921, 11.06118924484871, 11.26125944260169, 11.62872864165087, 12.34378622474458, 12.63655605428560, 13.08560551291501

Graph of the ZZ-function along the critical line