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Properties

Label 2-3192-3192.1637-c0-0-7
Degree 22
Conductor 31923192
Sign 0.939−0.342i0.939 - 0.342i
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  + (0.766 + 0.642i)2-s + (0.642 − 0.766i)3-s + (0.173 + 0.984i)4-s + (1.50 + 0.266i)5-s + (0.984 − 0.173i)6-s + (−0.5 − 0.866i)7-s + (−0.500 + 0.866i)8-s + (−0.173 − 0.984i)9-s + (0.984 + 1.17i)10-s + (0.866 + 0.500i)12-s + (−0.118 + 0.326i)13-s + (0.173 − 0.984i)14-s + (1.17 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (0.500 − 0.866i)18-s + (−0.642 + 0.766i)19-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)2-s + (0.642 − 0.766i)3-s + (0.173 + 0.984i)4-s + (1.50 + 0.266i)5-s + (0.984 − 0.173i)6-s + (−0.5 − 0.866i)7-s + (−0.500 + 0.866i)8-s + (−0.173 − 0.984i)9-s + (0.984 + 1.17i)10-s + (0.866 + 0.500i)12-s + (−0.118 + 0.326i)13-s + (0.173 − 0.984i)14-s + (1.17 − 0.984i)15-s + (−0.939 + 0.342i)16-s + (0.500 − 0.866i)18-s + (−0.642 + 0.766i)19-s + ⋯

Functional equation

Λ(s)=(3192s/2 ΓC(s) L(s)=((0.939−0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3192s/2 ΓC(s) L(s)=((0.939−0.342i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31923192    =    23⋅3⋅7⋅192^{3} \cdot 3 \cdot 7 \cdot 19
Sign: 0.939−0.342i0.939 - 0.342i
Analytic conductor: 1.593011.59301
Root analytic conductor: 1.262141.26214
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3192(1637,⋅)\chi_{3192} (1637, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3192, ( :0), 0.939−0.342i)(2,\ 3192,\ (\ :0),\ 0.939 - 0.342i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 2.8117856222.811785622
L(12)L(\frac12) ≈\approx 2.8117856222.811785622
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+(−0.766−0.642i)T 1 + (-0.766 - 0.642i)T
3 1+(−0.642+0.766i)T 1 + (-0.642 + 0.766i)T
7 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
19 1+(0.642−0.766i)T 1 + (0.642 - 0.766i)T
good5 1+(−1.50−0.266i)T+(0.939+0.342i)T2 1 + (-1.50 - 0.266i)T + (0.939 + 0.342i)T^{2}
11 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
13 1+(0.118−0.326i)T+(−0.766−0.642i)T2 1 + (0.118 - 0.326i)T + (-0.766 - 0.642i)T^{2}
17 1+(0.173+0.984i)T2 1 + (0.173 + 0.984i)T^{2}
23 1+(−1.26+0.223i)T+(0.939−0.342i)T2 1 + (-1.26 + 0.223i)T + (0.939 - 0.342i)T^{2}
29 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
31 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
37 1+T2 1 + T^{2}
41 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
43 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
47 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
53 1+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
59 1+(0.524+0.439i)T+(0.173+0.984i)T2 1 + (0.524 + 0.439i)T + (0.173 + 0.984i)T^{2}
61 1+(−0.118−0.673i)T+(−0.939+0.342i)T2 1 + (-0.118 - 0.673i)T + (-0.939 + 0.342i)T^{2}
67 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
71 1+(−0.326+1.85i)T+(−0.939−0.342i)T2 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2}
73 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
79 1+(0.592+1.62i)T+(−0.766+0.642i)T2 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2}
83 1+(1.62−0.939i)T+(0.5−0.866i)T2 1 + (1.62 - 0.939i)T + (0.5 - 0.866i)T^{2}
89 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)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

−8.859295475605796413847640324422, −7.900489967927115779190354845658, −7.18780932890153139609785933188, −6.51828935494866110683378979845, −6.19241149994916940387049266903, −5.24841271232930080551240095211, −4.18992108939880932943756093271, −3.26099658242100543389622156632, −2.51309619547779340185240810182, −1.54837548671595898967083528712, 1.60589006947095946729717713245, 2.60127565340429153397172884975, 2.90478187186551185716107912363, 4.11187673109320410880182398467, 5.09402192467690183025455248771, 5.44042058576542142959957938599, 6.23484028392094730539457824775, 7.06335125046196356972403798844, 8.564519644605829795806595742743, 9.025573795553814602669429538789

Graph of the ZZ-function along the critical line