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Properties

Label 2-3549-21.11-c0-0-1
Degree 22
Conductor 35493549
Sign −0.832−0.553i-0.832 - 0.553i
Analytic cond. 1.771181.77118
Root an. cond. 1.330851.33085
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  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + 7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + (−0.499 − 0.866i)16-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (−0.5 + 0.866i)28-s + (−1 + 1.73i)31-s + 0.999·36-s + (0.5 + 0.866i)37-s − 43-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + 7-s + (−0.499 − 0.866i)9-s + (−0.499 − 0.866i)12-s + (−0.499 − 0.866i)16-s + (0.5 + 0.866i)19-s + (−0.5 + 0.866i)21-s + (−0.5 + 0.866i)25-s + 0.999·27-s + (−0.5 + 0.866i)28-s + (−1 + 1.73i)31-s + 0.999·36-s + (0.5 + 0.866i)37-s − 43-s + ⋯

Functional equation

Λ(s)=(3549s/2 ΓC(s) L(s)=((−0.832−0.553i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3549s/2 ΓC(s) L(s)=((−0.832−0.553i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 35493549    =    3⋅7⋅1323 \cdot 7 \cdot 13^{2}
Sign: −0.832−0.553i-0.832 - 0.553i
Analytic conductor: 1.771181.77118
Root analytic conductor: 1.330851.33085
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3549(1691,⋅)\chi_{3549} (1691, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3549, ( :0), −0.832−0.553i)(2,\ 3549,\ (\ :0),\ -0.832 - 0.553i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.85069116520.8506911652
L(12)L(\frac12) ≈\approx 0.85069116520.8506911652
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+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
7 1−T 1 - T
13 1 1
good2 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
5 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
11 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
17 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
19 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
23 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
29 1−T2 1 - T^{2}
31 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
37 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1+T+T2 1 + T + T^{2}
47 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
53 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
59 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
61 1+(−0.5−0.866i)T+(−0.5+0.866i)T2 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}
67 1+(1−1.73i)T+(−0.5−0.866i)T2 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}
71 1−T2 1 - T^{2}
73 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
79 1+(1+1.73i)T+(−0.5+0.866i)T2 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2}
83 1−T2 1 - T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1+T+T2 1 + T + 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

−8.922293409778590822462179376171, −8.465393136997674588035781937818, −7.65193959280154279638913303689, −6.95505763591245309022701211096, −5.76529162023349609869879688847, −5.14746419231732647853936141731, −4.48959681668215419117772241296, −3.70398828818316253035734409946, −3.01949551390773177709447818940, −1.48809322494757712281314394824, 0.56458621881697528772267538142, 1.68322333978594262498757629335, 2.43056504684672828421140871339, 4.03528168321679521148877699674, 4.84628486392782537195623458802, 5.47270839511334773130692100095, 6.08764886848978946299411249578, 6.93405538086859487749985995299, 7.72904451933429480594957306273, 8.313973223058416151096288384942

Graph of the ZZ-function along the critical line