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Properties

Label 2-825-1.1-c3-0-73
Degree 22
Conductor 825825
Sign −1-1
Analytic cond. 48.676548.6765
Root an. cond. 6.976866.97686
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 − 8·4-s − 2·7-s + 9·9-s − 11·11-s − 24·12-s + 22·13-s + 64·16-s − 72·17-s + 122·19-s − 6·21-s − 72·23-s + 27·27-s + 16·28-s + 96·29-s − 112·31-s − 33·33-s − 72·36-s − 266·37-s + 66·39-s − 96·41-s + 382·43-s + 88·44-s − 360·47-s + 192·48-s − 339·49-s − 216·51-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.107·7-s + 1/3·9-s − 0.301·11-s − 0.577·12-s + 0.469·13-s + 16-s − 1.02·17-s + 1.47·19-s − 0.0623·21-s − 0.652·23-s + 0.192·27-s + 0.107·28-s + 0.614·29-s − 0.648·31-s − 0.174·33-s − 1/3·36-s − 1.18·37-s + 0.270·39-s − 0.365·41-s + 1.35·43-s + 0.301·44-s − 1.11·47-s + 0.577·48-s − 0.988·49-s − 0.593·51-s + ⋯

Functional equation

Λ(s)=(825s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(825s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 825825    =    3⋅52⋅113 \cdot 5^{2} \cdot 11
Sign: −1-1
Analytic conductor: 48.676548.6765
Root analytic conductor: 6.976866.97686
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 825, ( :3/2), −1)(2,\ 825,\ (\ :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)
bad3 1−pT 1 - p T
5 1 1
11 1+pT 1 + p T
good2 1+p3T2 1 + p^{3} T^{2}
7 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
13 1−22T+p3T2 1 - 22 T + p^{3} T^{2}
17 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
19 1−122T+p3T2 1 - 122 T + p^{3} T^{2}
23 1+72T+p3T2 1 + 72 T + p^{3} T^{2}
29 1−96T+p3T2 1 - 96 T + p^{3} T^{2}
31 1+112T+p3T2 1 + 112 T + p^{3} T^{2}
37 1+266T+p3T2 1 + 266 T + p^{3} T^{2}
41 1+96T+p3T2 1 + 96 T + p^{3} T^{2}
43 1−382T+p3T2 1 - 382 T + p^{3} T^{2}
47 1+360T+p3T2 1 + 360 T + p^{3} T^{2}
53 1+6pT+p3T2 1 + 6 p T + p^{3} T^{2}
59 1−660T+p3T2 1 - 660 T + p^{3} T^{2}
61 1+430T+p3T2 1 + 430 T + p^{3} T^{2}
67 1+380T+p3T2 1 + 380 T + p^{3} T^{2}
71 1−168T+p3T2 1 - 168 T + p^{3} T^{2}
73 1+218T+p3T2 1 + 218 T + p^{3} T^{2}
79 1+706T+p3T2 1 + 706 T + p^{3} T^{2}
83 1+1068T+p3T2 1 + 1068 T + p^{3} T^{2}
89 1+6T+p3T2 1 + 6 T + p^{3} T^{2}
97 1+686T+p3T2 1 + 686 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

−9.363252259449192128624865426005, −8.610521330296451039658275045690, −7.928454194132104779784890868792, −6.94268162317599509189957641743, −5.74437026653176259668032328609, −4.81776315307113769974819199589, −3.87496281931032794610050345834, −2.95865112366885103391089029523, −1.46089361123687494602973421762, 0, 1.46089361123687494602973421762, 2.95865112366885103391089029523, 3.87496281931032794610050345834, 4.81776315307113769974819199589, 5.74437026653176259668032328609, 6.94268162317599509189957641743, 7.928454194132104779784890868792, 8.610521330296451039658275045690, 9.363252259449192128624865426005

Graph of the ZZ-function along the critical line