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Properties

Label 2-226512-1.1-c1-0-123
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  + 2·5-s − 3·7-s + 13-s + 19-s + 4·23-s − 25-s − 8·29-s − 9·31-s − 6·35-s + 9·37-s + 2·41-s − 4·43-s + 2·47-s + 2·49-s − 2·53-s + 6·59-s + 11·61-s + 2·65-s + 15·67-s + 6·71-s − 9·73-s + 79-s − 8·89-s − 3·91-s + 2·95-s + 7·97-s + 101-s + ⋯
L(s)  = 1  + 0.894·5-s − 1.13·7-s + 0.277·13-s + 0.229·19-s + 0.834·23-s − 1/5·25-s − 1.48·29-s − 1.61·31-s − 1.01·35-s + 1.47·37-s + 0.312·41-s − 0.609·43-s + 0.291·47-s + 2/7·49-s − 0.274·53-s + 0.781·59-s + 1.40·61-s + 0.248·65-s + 1.83·67-s + 0.712·71-s − 1.05·73-s + 0.112·79-s − 0.847·89-s − 0.314·91-s + 0.205·95-s + 0.710·97-s + 0.0995·101-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 1−2T+pT2 1 - 2 T + p T^{2}
7 1+3T+pT2 1 + 3 T + p T^{2}
17 1+pT2 1 + p T^{2}
19 1−T+pT2 1 - T + p T^{2}
23 1−4T+pT2 1 - 4 T + p T^{2}
29 1+8T+pT2 1 + 8 T + p T^{2}
31 1+9T+pT2 1 + 9 T + p T^{2}
37 1−9T+pT2 1 - 9 T + p T^{2}
41 1−2T+pT2 1 - 2 T + p T^{2}
43 1+4T+pT2 1 + 4 T + p T^{2}
47 1−2T+pT2 1 - 2 T + p T^{2}
53 1+2T+pT2 1 + 2 T + p T^{2}
59 1−6T+pT2 1 - 6 T + p T^{2}
61 1−11T+pT2 1 - 11 T + p T^{2}
67 1−15T+pT2 1 - 15 T + p T^{2}
71 1−6T+pT2 1 - 6 T + p T^{2}
73 1+9T+pT2 1 + 9 T + p T^{2}
79 1−T+pT2 1 - T + p T^{2}
83 1+pT2 1 + p T^{2}
89 1+8T+pT2 1 + 8 T + p T^{2}
97 1−7T+pT2 1 - 7 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

−13.05491680304926, −12.91872304527535, −12.52094289364635, −11.63421025927391, −11.29114744205481, −10.93716710220349, −10.21934484017617, −9.794587782708962, −9.557152909591531, −9.034246707829606, −8.711541212122672, −7.853010489290893, −7.502174851429844, −6.838790239539393, −6.529728819577427, −5.948269739647687, −5.475037972340854, −5.224438526913785, −4.321620262251457, −3.666511241986788, −3.450016926540343, −2.604339974998391, −2.211884219756977, −1.525418021852300, −0.7912401197147888, 0, 0.7912401197147888, 1.525418021852300, 2.211884219756977, 2.604339974998391, 3.450016926540343, 3.666511241986788, 4.321620262251457, 5.224438526913785, 5.475037972340854, 5.948269739647687, 6.529728819577427, 6.838790239539393, 7.502174851429844, 7.853010489290893, 8.711541212122672, 9.034246707829606, 9.557152909591531, 9.794587782708962, 10.21934484017617, 10.93716710220349, 11.29114744205481, 11.63421025927391, 12.52094289364635, 12.91872304527535, 13.05491680304926

Graph of the ZZ-function along the critical line