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Properties

Label 2-315-1.1-c3-0-29
Degree 22
Conductor 315315
Sign −1-1
Analytic cond. 18.585618.5856
Root an. cond. 4.311104.31110
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.82·2-s + 6.65·4-s − 5·5-s − 7·7-s − 5.14·8-s − 19.1·10-s − 48.5·11-s − 43.6·13-s − 26.7·14-s − 72.9·16-s + 67.6·17-s − 93.2·19-s − 33.2·20-s − 185.·22-s + 104.·23-s + 25·25-s − 167.·26-s − 46.5·28-s + 58.7·29-s − 9.08·31-s − 238.·32-s + 259.·34-s + 35·35-s − 252.·37-s − 357.·38-s + 25.7·40-s − 276.·41-s + ⋯
L(s)  = 1  + 1.35·2-s + 0.832·4-s − 0.447·5-s − 0.377·7-s − 0.227·8-s − 0.605·10-s − 1.33·11-s − 0.931·13-s − 0.511·14-s − 1.13·16-s + 0.965·17-s − 1.12·19-s − 0.372·20-s − 1.80·22-s + 0.944·23-s + 0.200·25-s − 1.26·26-s − 0.314·28-s + 0.376·29-s − 0.0526·31-s − 1.31·32-s + 1.30·34-s + 0.169·35-s − 1.12·37-s − 1.52·38-s + 0.101·40-s − 1.05·41-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 18.585618.5856
Root analytic conductor: 4.311104.31110
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 315, ( :3/2), −1)(2,\ 315,\ (\ :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 1
5 1+5T 1 + 5T
7 1+7T 1 + 7T
good2 1−3.82T+8T2 1 - 3.82T + 8T^{2}
11 1+48.5T+1.33e3T2 1 + 48.5T + 1.33e3T^{2}
13 1+43.6T+2.19e3T2 1 + 43.6T + 2.19e3T^{2}
17 1−67.6T+4.91e3T2 1 - 67.6T + 4.91e3T^{2}
19 1+93.2T+6.85e3T2 1 + 93.2T + 6.85e3T^{2}
23 1−104.T+1.21e4T2 1 - 104.T + 1.21e4T^{2}
29 1−58.7T+2.43e4T2 1 - 58.7T + 2.43e4T^{2}
31 1+9.08T+2.97e4T2 1 + 9.08T + 2.97e4T^{2}
37 1+252.T+5.06e4T2 1 + 252.T + 5.06e4T^{2}
41 1+276.T+6.89e4T2 1 + 276.T + 6.89e4T^{2}
43 1+92.6T+7.95e4T2 1 + 92.6T + 7.95e4T^{2}
47 1−582.T+1.03e5T2 1 - 582.T + 1.03e5T^{2}
53 1+623.T+1.48e5T2 1 + 623.T + 1.48e5T^{2}
59 1−524.T+2.05e5T2 1 - 524.T + 2.05e5T^{2}
61 1+352.T+2.26e5T2 1 + 352.T + 2.26e5T^{2}
67 1+736.T+3.00e5T2 1 + 736.T + 3.00e5T^{2}
71 1−492.T+3.57e5T2 1 - 492.T + 3.57e5T^{2}
73 1−1.16e3T+3.89e5T2 1 - 1.16e3T + 3.89e5T^{2}
79 1+872.T+4.93e5T2 1 + 872.T + 4.93e5T^{2}
83 1−529.T+5.71e5T2 1 - 529.T + 5.71e5T^{2}
89 1−385.T+7.04e5T2 1 - 385.T + 7.04e5T^{2}
97 1+463.T+9.12e5T2 1 + 463.T + 9.12e5T^{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

−10.94134673100186775562682202165, −10.07405441550256819965094553309, −8.793958841921703801403813658371, −7.64082229078550582013608267198, −6.64470175715975278717112869722, −5.42455824634066542670465415236, −4.74262187066406129836835954238, −3.48151172056530155917647529141, −2.53407874209039872863235914006, 0, 2.53407874209039872863235914006, 3.48151172056530155917647529141, 4.74262187066406129836835954238, 5.42455824634066542670465415236, 6.64470175715975278717112869722, 7.64082229078550582013608267198, 8.793958841921703801403813658371, 10.07405441550256819965094553309, 10.94134673100186775562682202165

Graph of the ZZ-function along the critical line