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Properties

Label 2-525-1.1-c1-0-18
Degree 22
Conductor 525525
Sign −1-1
Analytic cond. 4.192144.19214
Root an. cond. 2.047472.04747
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-s − 3-s − 4-s − 6-s + 7-s − 3·8-s + 9-s − 6·11-s + 12-s + 2·13-s + 14-s − 16-s − 4·17-s + 18-s − 6·19-s − 21-s − 6·22-s + 3·24-s + 2·26-s − 27-s − 28-s − 2·29-s − 10·31-s + 5·32-s + 6·33-s − 4·34-s − 36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s + 0.235·18-s − 1.37·19-s − 0.218·21-s − 1.27·22-s + 0.612·24-s + 0.392·26-s − 0.192·27-s − 0.188·28-s − 0.371·29-s − 1.79·31-s + 0.883·32-s + 1.04·33-s − 0.685·34-s − 1/6·36-s + ⋯

Functional equation

Λ(s)=(525s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(525s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, 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: 4.192144.19214
Root analytic conductor: 2.047472.04747
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 525, ( :1/2), −1)(2,\ 525,\ (\ :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)
bad3 1+T 1 + T
5 1 1
7 1−T 1 - T
good2 1−T+pT2 1 - T + p T^{2}
11 1+6T+pT2 1 + 6 T + p T^{2}
13 1−2T+pT2 1 - 2 T + p T^{2}
17 1+4T+pT2 1 + 4 T + p T^{2}
19 1+6T+pT2 1 + 6 T + p T^{2}
23 1+pT2 1 + p T^{2}
29 1+2T+pT2 1 + 2 T + p T^{2}
31 1+10T+pT2 1 + 10 T + p T^{2}
37 1−4T+pT2 1 - 4 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+pT2 1 + p T^{2}
53 1+6T+pT2 1 + 6 T + p T^{2}
59 1+8T+pT2 1 + 8 T + p T^{2}
61 1+2T+pT2 1 + 2 T + p T^{2}
67 1−16T+pT2 1 - 16 T + p T^{2}
71 1−10T+pT2 1 - 10 T + p T^{2}
73 1−6T+pT2 1 - 6 T + p T^{2}
79 1−4T+pT2 1 - 4 T + p T^{2}
83 1+8T+pT2 1 + 8 T + p T^{2}
89 1−6T+pT2 1 - 6 T + p T^{2}
97 1−2T+pT2 1 - 2 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

−10.87163300470130379010441843726, −9.564382086134303299667080987913, −8.593374216240694518737306786941, −7.73714417281129970127485027333, −6.42340127852064511914793730163, −5.52967641623344487457447591913, −4.80452837700174952413082108451, −3.85581516222527937213198033254, −2.35069263133539802699895815583, 0, 2.35069263133539802699895815583, 3.85581516222527937213198033254, 4.80452837700174952413082108451, 5.52967641623344487457447591913, 6.42340127852064511914793730163, 7.73714417281129970127485027333, 8.593374216240694518737306786941, 9.564382086134303299667080987913, 10.87163300470130379010441843726

Graph of the ZZ-function along the critical line