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Properties

Label 2-3192-3192.2621-c0-0-11
Degree 22
Conductor 31923192
Sign −0.954+0.296i-0.954 + 0.296i
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  + (0.866 − 0.5i)2-s + (−0.342 − 0.939i)3-s + (0.499 − 0.866i)4-s + (−0.766 − 0.642i)6-s + (0.173 − 0.984i)7-s − 0.999i·8-s + (−0.766 + 0.642i)9-s + (−0.984 − 0.173i)12-s − 0.347i·13-s + (−0.342 − 0.939i)14-s + (−0.5 − 0.866i)16-s + (0.939 − 1.62i)17-s + (−0.342 + 0.939i)18-s + (−0.866 + 0.5i)19-s + (−0.984 + 0.173i)21-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.342 − 0.939i)3-s + (0.499 − 0.866i)4-s + (−0.766 − 0.642i)6-s + (0.173 − 0.984i)7-s − 0.999i·8-s + (−0.766 + 0.642i)9-s + (−0.984 − 0.173i)12-s − 0.347i·13-s + (−0.342 − 0.939i)14-s + (−0.5 − 0.866i)16-s + (0.939 − 1.62i)17-s + (−0.342 + 0.939i)18-s + (−0.866 + 0.5i)19-s + (−0.984 + 0.173i)21-s + ⋯

Functional equation

Λ(s)=(3192s/2 ΓC(s) L(s)=((−0.954+0.296i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3192s/2 ΓC(s) L(s)=((−0.954+0.296i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 31923192    =    23⋅3⋅7⋅192^{3} \cdot 3 \cdot 7 \cdot 19
Sign: −0.954+0.296i-0.954 + 0.296i
Analytic conductor: 1.593011.59301
Root analytic conductor: 1.262141.26214
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3192(2621,⋅)\chi_{3192} (2621, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3192, ( :0), −0.954+0.296i)(2,\ 3192,\ (\ :0),\ -0.954 + 0.296i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.7671813581.767181358
L(12)L(\frac12) ≈\approx 1.7671813581.767181358
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+(−0.866+0.5i)T 1 + (-0.866 + 0.5i)T
3 1+(0.342+0.939i)T 1 + (0.342 + 0.939i)T
7 1+(−0.173+0.984i)T 1 + (-0.173 + 0.984i)T
19 1+(0.866−0.5i)T 1 + (0.866 - 0.5i)T
good5 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}
13 1+0.347iT−T2 1 + 0.347iT - 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}
23 1+(−0.592+0.342i)T+(0.5−0.866i)T2 1 + (-0.592 + 0.342i)T + (0.5 - 0.866i)T^{2}
29 1−1.53iT−T2 1 - 1.53iT - T^{2}
31 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
37 1+(−0.866−1.5i)T+(−0.5+0.866i)T2 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2}
41 1−T2 1 - T^{2}
43 1−T2 1 - T^{2}
47 1+(0.5+0.866i)T+(−0.5+0.866i)T2 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2}
53 1+(1.62+0.939i)T+(0.5+0.866i)T2 1 + (1.62 + 0.939i)T + (0.5 + 0.866i)T^{2}
59 1+(−0.342+0.592i)T+(−0.5−0.866i)T2 1 + (-0.342 + 0.592i)T + (-0.5 - 0.866i)T^{2}
61 1+(−0.5+0.866i)T2 1 + (-0.5 + 0.866i)T^{2}
67 1+(−0.642+1.11i)T+(−0.5−0.866i)T2 1 + (-0.642 + 1.11i)T + (-0.5 - 0.866i)T^{2}
71 1+T2 1 + T^{2}
73 1+(−1.70−0.984i)T+(0.5+0.866i)T2 1 + (-1.70 - 0.984i)T + (0.5 + 0.866i)T^{2}
79 1+(0.5−0.866i)T2 1 + (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+T2 1 + 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

−8.254780545097994750962261716935, −7.54627002111252365473592591753, −6.85865342049085588823098523019, −6.34359703023976138771812679321, −5.12563760783831688890445186903, −4.98928676468412727519427301230, −3.60773897327391233123917194239, −2.93921234166794158473293545477, −1.75073290668055382642974212076, −0.845989587367407715093163783675, 2.08906710719566979541769130331, 3.00639047977822814268995809553, 4.03479657514882606983227935792, 4.47572824154237059185446252359, 5.49866311781260380780542466930, 5.99592619726116186175617971009, 6.49496718655254726117190651521, 7.83630977914682557326175293420, 8.311578903065941209930163194144, 9.126404926942034392960838938061

Graph of the ZZ-function along the critical line