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Properties

Label 2-525-1.1-c3-0-43
Degree 22
Conductor 525525
Sign −1-1
Analytic cond. 30.976030.9760
Root an. cond. 5.565605.56560
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 − 7·7-s + 9·9-s + 42·11-s − 24·12-s − 20·13-s + 64·16-s − 66·17-s + 38·19-s − 21·21-s − 12·23-s + 27·27-s + 56·28-s − 258·29-s + 146·31-s + 126·33-s − 72·36-s − 434·37-s − 60·39-s − 282·41-s − 20·43-s − 336·44-s + 72·47-s + 192·48-s + 49·49-s − 198·51-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s + 1.15·11-s − 0.577·12-s − 0.426·13-s + 16-s − 0.941·17-s + 0.458·19-s − 0.218·21-s − 0.108·23-s + 0.192·27-s + 0.377·28-s − 1.65·29-s + 0.845·31-s + 0.664·33-s − 1/3·36-s − 1.92·37-s − 0.246·39-s − 1.07·41-s − 0.0709·43-s − 1.15·44-s + 0.223·47-s + 0.577·48-s + 1/7·49-s − 0.543·51-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 525525    =    3⋅52⋅73 \cdot 5^{2} \cdot 7
Sign: −1-1
Analytic conductor: 30.976030.9760
Root analytic conductor: 5.565605.56560
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 525, ( :3/2), −1)(2,\ 525,\ (\ :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
7 1+pT 1 + p T
good2 1+p3T2 1 + p^{3} T^{2}
11 1−42T+p3T2 1 - 42 T + p^{3} T^{2}
13 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
17 1+66T+p3T2 1 + 66 T + p^{3} T^{2}
19 1−2pT+p3T2 1 - 2 p T + p^{3} T^{2}
23 1+12T+p3T2 1 + 12 T + p^{3} T^{2}
29 1+258T+p3T2 1 + 258 T + p^{3} T^{2}
31 1−146T+p3T2 1 - 146 T + p^{3} T^{2}
37 1+434T+p3T2 1 + 434 T + p^{3} T^{2}
41 1+282T+p3T2 1 + 282 T + p^{3} T^{2}
43 1+20T+p3T2 1 + 20 T + p^{3} T^{2}
47 1−72T+p3T2 1 - 72 T + p^{3} T^{2}
53 1+336T+p3T2 1 + 336 T + p^{3} T^{2}
59 1+360T+p3T2 1 + 360 T + p^{3} T^{2}
61 1+682T+p3T2 1 + 682 T + p^{3} T^{2}
67 1+812T+p3T2 1 + 812 T + p^{3} T^{2}
71 1−810T+p3T2 1 - 810 T + p^{3} T^{2}
73 1−124T+p3T2 1 - 124 T + p^{3} T^{2}
79 1−1136T+p3T2 1 - 1136 T + p^{3} T^{2}
83 1+156T+p3T2 1 + 156 T + p^{3} T^{2}
89 1+1038T+p3T2 1 + 1038 T + p^{3} T^{2}
97 1+1208T+p3T2 1 + 1208 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

−9.660480682934644811494424762636, −9.198263605310083470927787310436, −8.446844434691486110679714658395, −7.36657299145429705693745425736, −6.38185322371920146910964906708, −5.10879224059252464092674021945, −4.10762185843126572820557107934, −3.26339512374771722020879159902, −1.63394024708422573519915901457, 0, 1.63394024708422573519915901457, 3.26339512374771722020879159902, 4.10762185843126572820557107934, 5.10879224059252464092674021945, 6.38185322371920146910964906708, 7.36657299145429705693745425736, 8.446844434691486110679714658395, 9.198263605310083470927787310436, 9.660480682934644811494424762636

Graph of the ZZ-function along the critical line