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Properties

Label 2-3192-3192.293-c0-0-0
Degree 22
Conductor 31923192
Sign −0.0977−0.995i-0.0977 - 0.995i
Analytic cond. 1.593011.59301
Root an. cond. 1.262141.26214
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)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−1.5 − 0.866i)5-s + (0.499 + 0.866i)6-s − 7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 − 0.866i)10-s − 0.999·12-s + (−1.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−1.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + 0.999·18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (−1.5 − 0.866i)5-s + (0.499 + 0.866i)6-s − 7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (1.5 − 0.866i)10-s − 0.999·12-s + (−1.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−1.5 + 0.866i)15-s + (−0.5 + 0.866i)16-s + 0.999·18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

Λ(s)=(3192s/2 ΓC(s) L(s)=((−0.0977−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3192s/2 ΓC(s) L(s)=((−0.0977−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0977 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31923192    =    23⋅3⋅7⋅192^{3} \cdot 3 \cdot 7 \cdot 19
Sign: −0.0977−0.995i-0.0977 - 0.995i
Analytic conductor: 1.593011.59301
Root analytic conductor: 1.262141.26214
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3192(293,⋅)\chi_{3192} (293, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3192, ( :0), −0.0977−0.995i)(2,\ 3192,\ (\ :0),\ -0.0977 - 0.995i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.24013334710.2401333471
L(12)L(\frac12) ≈\approx 0.24013334710.2401333471
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)
bad2 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
3 1+(−0.5+0.866i)T 1 + (-0.5 + 0.866i)T
7 1+T 1 + T
19 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
good5 1+(1.5+0.866i)T+(0.5+0.866i)T2 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2}
11 1+T2 1 + T^{2}
13 1+(1.5−0.866i)T+(0.5−0.866i)T2 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2}
17 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
23 1+(−1.5+0.866i)T+(0.5−0.866i)T2 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2}
29 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
31 1+T2 1 + T^{2}
37 1+T2 1 + T^{2}
41 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
43 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)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)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-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+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
71 1+(0.5−0.866i)T+(−0.5−0.866i)T2 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2}
73 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
79 1+(0.5+0.866i)T2 1 + (0.5 + 0.866i)T^{2}
83 1−1.73iT−T2 1 - 1.73iT - T^{2}
89 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
97 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)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

−8.810452934543821562855604267071, −8.276895623343262516528129453027, −7.48641505484651745734630523063, −7.03876555467383651228061988423, −6.44896327108019872231178810125, −5.30493626042269676763051414738, −4.45627709155130449637918834369, −3.68691414023360777278620261986, −2.44232183434193954111126980024, −0.992901999810016422984879924284, 0.19881365378083499981527756176, 2.58276311525575251510711829252, 3.00475726026346228304954978557, 3.61294623114665113948270869397, 4.44840249652701646765990762909, 5.20915196316881428722451522688, 6.82119189410022224671334497687, 7.43844053830505976544585260267, 7.927657198259334592580998090980, 8.902987144442019390671074173631

Graph of the ZZ-function along the critical line