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Properties

Label 2-3645-135.14-c0-0-0
Degree 22
Conductor 36453645
Sign 0.957+0.286i0.957 + 0.286i
Analytic cond. 1.819091.81909
Root an. cond. 1.348731.34873
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.266 − 1.50i)2-s + (−1.26 + 0.460i)4-s + (−0.766 − 0.642i)5-s + (0.266 + 0.460i)8-s + (−0.766 + 1.32i)10-s + (−0.407 + 0.342i)16-s + (−0.939 + 1.62i)17-s + (0.939 + 1.62i)19-s + (1.26 + 0.460i)20-s + (0.326 − 0.118i)23-s + (0.173 + 0.984i)25-s + (−1.43 + 0.524i)31-s + (1.03 + 0.866i)32-s + (2.70 + 0.984i)34-s + (2.20 − 1.85i)38-s + ⋯
L(s)  = 1  + (−0.266 − 1.50i)2-s + (−1.26 + 0.460i)4-s + (−0.766 − 0.642i)5-s + (0.266 + 0.460i)8-s + (−0.766 + 1.32i)10-s + (−0.407 + 0.342i)16-s + (−0.939 + 1.62i)17-s + (0.939 + 1.62i)19-s + (1.26 + 0.460i)20-s + (0.326 − 0.118i)23-s + (0.173 + 0.984i)25-s + (−1.43 + 0.524i)31-s + (1.03 + 0.866i)32-s + (2.70 + 0.984i)34-s + (2.20 − 1.85i)38-s + ⋯

Functional equation

Λ(s)=(3645s/2 ΓC(s) L(s)=((0.957+0.286i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3645s/2 ΓC(s) L(s)=((0.957+0.286i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3645 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.286i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 36453645    =    36⋅53^{6} \cdot 5
Sign: 0.957+0.286i0.957 + 0.286i
Analytic conductor: 1.819091.81909
Root analytic conductor: 1.348731.34873
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3645(2834,⋅)\chi_{3645} (2834, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3645, ( :0), 0.957+0.286i)(2,\ 3645,\ (\ :0),\ 0.957 + 0.286i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.52790720770.5279072077
L(12)L(\frac12) ≈\approx 0.52790720770.5279072077
L(1)L(1) 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+(0.766+0.642i)T 1 + (0.766 + 0.642i)T
good2 1+(0.266+1.50i)T+(−0.939+0.342i)T2 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2}
7 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
11 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
13 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
17 1+(0.939−1.62i)T+(−0.5−0.866i)T2 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2}
19 1+(−0.939−1.62i)T+(−0.5+0.866i)T2 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2}
23 1+(−0.326+0.118i)T+(0.766−0.642i)T2 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2}
29 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
31 1+(1.43−0.524i)T+(0.766−0.642i)T2 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2}
37 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
41 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
43 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
47 1+(0.939+0.342i)T+(0.766+0.642i)T2 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2}
53 1+0.347T+T2 1 + 0.347T + T^{2}
59 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
61 1+(1.43+0.524i)T+(0.766+0.642i)T2 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2}
67 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
71 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
73 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
79 1+(−0.0603−0.342i)T+(−0.939+0.342i)T2 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2}
83 1+(−0.326−1.85i)T+(−0.939+0.342i)T2 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)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

−8.814748185280411468423177075827, −8.244651075216763627430717990307, −7.50257302724548445337270521014, −6.43085355174188743833996786674, −5.48434789997848187884558554269, −4.49345806804556776517615087665, −3.77901751627453704275834267805, −3.29709405123511897189571428868, −1.94944584391619836303582294771, −1.27863513969111598989428988487, 0.35563376167135939988379420198, 2.48127708632949201211611388845, 3.30017500722899932065874918617, 4.58581335399057711969887590909, 5.00204055613944564378515554890, 5.99989380997831250081246181092, 6.81523832302034333083246950160, 7.34062680921395021789446877603, 7.57157848662839008671540637236, 8.724786707899090744216956524661

Graph of the ZZ-function along the critical line