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Properties

Label 2-3192-3192.293-c0-0-4
Degree 22
Conductor 31923192
Sign 0.0977+0.995i0.0977 + 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 + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + (−0.866 + 0.499i)6-s + 7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)10-s + 0.999i·12-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)14-s + (−0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + 0.999·18-s + (0.866 + 0.5i)19-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (0.866 + 0.5i)5-s + (−0.866 + 0.499i)6-s + 7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.866 − 0.499i)10-s + 0.999i·12-s + (−0.866 + 0.5i)13-s + (0.5 − 0.866i)14-s + (−0.499 − 0.866i)15-s + (−0.5 + 0.866i)16-s + 0.999·18-s + (0.866 + 0.5i)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+0.995i0.0977 + 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 1.4964904221.496490422
L(12)L(\frac12) ≈\approx 1.4964904221.496490422
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+(0.866+0.5i)T 1 + (0.866 + 0.5i)T
7 1−T 1 - T
19 1+(−0.866−0.5i)T 1 + (-0.866 - 0.5i)T
good5 1+(−0.866−0.5i)T+(0.5+0.866i)T2 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2}
11 1+T2 1 + T^{2}
13 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)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+(−0.866+1.5i)T+(−0.5−0.866i)T2 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2}
61 1+(−0.866−1.5i)T+(−0.5+0.866i)T2 1 + (-0.866 - 1.5i)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+iT−T2 1 + iT - 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

−8.861498761055730257390243871084, −7.82322309861756655983741660636, −6.92797996854415093887789552043, −6.31956014410785507253996313075, −5.31833714962903270830358321995, −5.06688982887023827080769456020, −4.11897346773153145507215001079, −2.71865252490534595979859692798, −2.04451999782988839908146878045, −1.12487064132438352961850515991, 1.17107950347365331402196183306, 2.71815908182325246368245657223, 3.87886769325986953374021911015, 4.85401896802809214113491674251, 5.32542584834150263108818304527, 5.56143716035259711039699030330, 6.75518535107221849194960828199, 7.31487290995291755200036353332, 8.159561332948364338982196930529, 9.172393271590976973098316937339

Graph of the ZZ-function along the critical line