↑

Properties

Label 2-7605-1.1-c1-0-218
Degree 22
Conductor 76057605
Sign −1-1
Analytic cond. 60.726260.7262
Root an. cond. 7.792707.79270
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 5-s + 5·7-s − 2·10-s − 2·11-s − 10·14-s − 4·16-s − 2·17-s + 2·20-s + 4·22-s − 6·23-s + 25-s + 10·28-s + 4·29-s − 7·31-s + 8·32-s + 4·34-s + 5·35-s − 2·37-s − 6·41-s + 43-s − 4·44-s + 12·46-s + 8·47-s + 18·49-s − 2·50-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.447·5-s + 1.88·7-s − 0.632·10-s − 0.603·11-s − 2.67·14-s − 16-s − 0.485·17-s + 0.447·20-s + 0.852·22-s − 1.25·23-s + 1/5·25-s + 1.88·28-s + 0.742·29-s − 1.25·31-s + 1.41·32-s + 0.685·34-s + 0.845·35-s − 0.328·37-s − 0.937·41-s + 0.152·43-s − 0.603·44-s + 1.76·46-s + 1.16·47-s + 18/7·49-s − 0.282·50-s + ⋯

Functional equation

Λ(s)=(7605s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7605s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 76057605    =    32⋅5⋅1323^{2} \cdot 5 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 60.726260.7262
Root analytic conductor: 7.792707.79270
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7605, ( :1/2), −1)(2,\ 7605,\ (\ :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 1
5 1−T 1 - T
13 1 1
good2 1+pT+pT2 1 + p T + p T^{2}
7 1−5T+pT2 1 - 5 T + p T^{2}
11 1+2T+pT2 1 + 2 T + p T^{2}
17 1+2T+pT2 1 + 2 T + p T^{2}
19 1+pT2 1 + p T^{2}
23 1+6T+pT2 1 + 6 T + p T^{2}
29 1−4T+pT2 1 - 4 T + p T^{2}
31 1+7T+pT2 1 + 7 T + p T^{2}
37 1+2T+pT2 1 + 2 T + p T^{2}
41 1+6T+pT2 1 + 6 T + p T^{2}
43 1−T+pT2 1 - T + p T^{2}
47 1−8T+pT2 1 - 8 T + p T^{2}
53 1−4T+pT2 1 - 4 T + p T^{2}
59 1+12T+pT2 1 + 12 T + p T^{2}
61 1+13T+pT2 1 + 13 T + p T^{2}
67 1+7T+pT2 1 + 7 T + p T^{2}
71 1+12T+pT2 1 + 12 T + p T^{2}
73 1−15T+pT2 1 - 15 T + p T^{2}
79 1−3T+pT2 1 - 3 T + p T^{2}
83 1+8T+pT2 1 + 8 T + p T^{2}
89 1+14T+pT2 1 + 14 T + p T^{2}
97 1+5T+pT2 1 + 5 T + p T^{2}
show more
show less
   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

−7.63191830349450506511065357207, −7.32424451922789077347092107222, −6.27120901501207590227932710169, −5.43210262742652919238429528027, −4.76304990812899016071365029911, −4.09272495499005085504981921440, −2.61280199081632125119869153918, −1.86315542563040252055657950846, −1.34475697320078698804067361472, 0, 1.34475697320078698804067361472, 1.86315542563040252055657950846, 2.61280199081632125119869153918, 4.09272495499005085504981921440, 4.76304990812899016071365029911, 5.43210262742652919238429528027, 6.27120901501207590227932710169, 7.32424451922789077347092107222, 7.63191830349450506511065357207

Graph of the ZZ-function along the critical line