↑

Properties

Label 2-3192-3192.2309-c0-0-2
Degree 22
Conductor 31923192
Sign −0.660−0.750i-0.660 - 0.750i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 + 0.984i)2-s + (0.984 − 0.173i)3-s + (−0.939 + 0.342i)4-s + (0.118 − 0.326i)5-s + (0.342 + 0.939i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.939 − 0.342i)9-s + (0.342 + 0.0603i)10-s + (−0.866 + 0.5i)12-s + (−1.20 + 1.43i)13-s + (−0.939 − 0.342i)14-s + (0.0603 − 0.342i)15-s + (0.766 − 0.642i)16-s + (0.5 + 0.866i)18-s + (−0.984 + 0.173i)19-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.984 − 0.173i)3-s + (−0.939 + 0.342i)4-s + (0.118 − 0.326i)5-s + (0.342 + 0.939i)6-s + (−0.5 + 0.866i)7-s + (−0.5 − 0.866i)8-s + (0.939 − 0.342i)9-s + (0.342 + 0.0603i)10-s + (−0.866 + 0.5i)12-s + (−1.20 + 1.43i)13-s + (−0.939 − 0.342i)14-s + (0.0603 − 0.342i)15-s + (0.766 − 0.642i)16-s + (0.5 + 0.866i)18-s + (−0.984 + 0.173i)19-s + ⋯

Functional equation

Λ(s)=(3192s/2 ΓC(s) L(s)=((−0.660−0.750i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3192s/2 ΓC(s) L(s)=((−0.660−0.750i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.660 - 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31923192    =    23⋅3⋅7⋅192^{3} \cdot 3 \cdot 7 \cdot 19
Sign: −0.660−0.750i-0.660 - 0.750i
Analytic conductor: 1.593011.59301
Root analytic conductor: 1.262141.26214
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3192(2309,⋅)\chi_{3192} (2309, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3192, ( :0), −0.660−0.750i)(2,\ 3192,\ (\ :0),\ -0.660 - 0.750i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.4726297631.472629763
L(12)L(\frac12) ≈\approx 1.4726297631.472629763
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+(−0.173−0.984i)T 1 + (-0.173 - 0.984i)T
3 1+(−0.984+0.173i)T 1 + (-0.984 + 0.173i)T
7 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
19 1+(0.984−0.173i)T 1 + (0.984 - 0.173i)T
good5 1+(−0.118+0.326i)T+(−0.766−0.642i)T2 1 + (-0.118 + 0.326i)T + (-0.766 - 0.642i)T^{2}
11 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
13 1+(1.20−1.43i)T+(−0.173−0.984i)T2 1 + (1.20 - 1.43i)T + (-0.173 - 0.984i)T^{2}
17 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
23 1+(−0.673−1.85i)T+(−0.766+0.642i)T2 1 + (-0.673 - 1.85i)T + (-0.766 + 0.642i)T^{2}
29 1+(0.939+0.342i)T2 1 + (0.939 + 0.342i)T^{2}
31 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
37 1+T2 1 + T^{2}
41 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
43 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
47 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
53 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
59 1+(−0.223−1.26i)T+(−0.939+0.342i)T2 1 + (-0.223 - 1.26i)T + (-0.939 + 0.342i)T^{2}
61 1+(−1.20+0.439i)T+(0.766−0.642i)T2 1 + (-1.20 + 0.439i)T + (0.766 - 0.642i)T^{2}
67 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
71 1+(−1.43−0.524i)T+(0.766+0.642i)T2 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2}
73 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
79 1+(1.11+1.32i)T+(−0.173+0.984i)T2 1 + (1.11 + 1.32i)T + (-0.173 + 0.984i)T^{2}
83 1+(1.32+0.766i)T+(0.5+0.866i)T2 1 + (1.32 + 0.766i)T + (0.5 + 0.866i)T^{2}
89 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
97 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
show more
show less
   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

−8.848031449440135108281484647592, −8.593330833242536752487287109614, −7.41472446371091251969584677248, −7.08222427936977934410203039327, −6.25494364046094153159548149822, −5.32399920924301241775982574473, −4.57894340193634725151414673070, −3.71677860404665465888075560897, −2.78026673657496065111108622509, −1.72328470186867443804892806328, 0.74598011841606071217000921999, 2.37853251376917190635343652448, 2.77369172208854216297896247568, 3.65518345923739576531744128793, 4.49028613397443116777618147695, 5.10836269818787684945121246636, 6.44218675897300695100281393218, 7.15244902828603055090120967831, 8.131917916663559748983320903394, 8.589406089571826328034013703402

Graph of the ZZ-function along the critical line