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Properties

Label 2-57e2-171.67-c0-0-1
Degree 22
Conductor 32493249
Sign 0.776+0.630i0.776 + 0.630i
Analytic cond. 1.621461.62146
Root an. cond. 1.273361.27336
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  + (0.642 + 0.766i)2-s + (−0.642 − 0.766i)3-s + (0.766 − 0.642i)5-s + (0.173 − 0.984i)6-s + 7-s + (0.866 − 0.499i)8-s + (−0.173 + 0.984i)9-s + (0.984 + 0.173i)10-s + (0.5 − 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.642 + 0.766i)14-s + (−0.984 − 0.173i)15-s + (0.939 + 0.342i)16-s + (−0.866 + 0.500i)18-s + (−0.642 − 0.766i)21-s + (0.984 − 0.173i)22-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (−0.642 − 0.766i)3-s + (0.766 − 0.642i)5-s + (0.173 − 0.984i)6-s + 7-s + (0.866 − 0.499i)8-s + (−0.173 + 0.984i)9-s + (0.984 + 0.173i)10-s + (0.5 − 0.866i)11-s + (−0.342 − 0.939i)13-s + (0.642 + 0.766i)14-s + (−0.984 − 0.173i)15-s + (0.939 + 0.342i)16-s + (−0.866 + 0.500i)18-s + (−0.642 − 0.766i)21-s + (0.984 − 0.173i)22-s + ⋯

Functional equation

Λ(s)=(3249s/2 ΓC(s) L(s)=((0.776+0.630i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(3249s/2 ΓC(s) L(s)=((0.776+0.630i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.776 + 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 32493249    =    32⋅1923^{2} \cdot 19^{2}
Sign: 0.776+0.630i0.776 + 0.630i
Analytic conductor: 1.621461.62146
Root analytic conductor: 1.273361.27336
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ3249(1777,⋅)\chi_{3249} (1777, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 3249, ( :0), 0.776+0.630i)(2,\ 3249,\ (\ :0),\ 0.776 + 0.630i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9768689851.976868985
L(12)L(\frac12) ≈\approx 1.9768689851.976868985
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)
bad3 1+(0.642+0.766i)T 1 + (0.642 + 0.766i)T
19 1 1
good2 1+(−0.642−0.766i)T+(−0.173+0.984i)T2 1 + (-0.642 - 0.766i)T + (-0.173 + 0.984i)T^{2}
5 1+(−0.766+0.642i)T+(0.173−0.984i)T2 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2}
7 1−T+T2 1 - T + T^{2}
11 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}
13 1+(0.342+0.939i)T+(−0.766+0.642i)T2 1 + (0.342 + 0.939i)T + (-0.766 + 0.642i)T^{2}
17 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
23 1+(0.173−0.984i)T+(−0.939−0.342i)T2 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}
29 1+(−0.342−0.939i)T+(−0.766+0.642i)T2 1 + (-0.342 - 0.939i)T + (-0.766 + 0.642i)T^{2}
31 1+(0.866−0.5i)T+(0.5−0.866i)T2 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2}
37 1−T2 1 - T^{2}
41 1+(0.984−0.173i)T+(0.939−0.342i)T2 1 + (0.984 - 0.173i)T + (0.939 - 0.342i)T^{2}
43 1+(−0.173−0.984i)T+(−0.939+0.342i)T2 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2}
47 1+(0.939−0.342i)T+(0.766−0.642i)T2 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2}
53 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
59 1+(0.342−0.939i)T+(−0.766−0.642i)T2 1 + (0.342 - 0.939i)T + (-0.766 - 0.642i)T^{2}
61 1+(0.766+0.642i)T+(0.173+0.984i)T2 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2}
67 1+(−0.642+0.766i)T+(−0.173−0.984i)T2 1 + (-0.642 + 0.766i)T + (-0.173 - 0.984i)T^{2}
71 1+(−0.766−0.642i)T2 1 + (-0.766 - 0.642i)T^{2}
73 1+(0.173−0.984i)T2 1 + (0.173 - 0.984i)T^{2}
79 1+(0.342−0.939i)T+(−0.766−0.642i)T2 1 + (0.342 - 0.939i)T + (-0.766 - 0.642i)T^{2}
83 1+T+T2 1 + T + T^{2}
89 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
97 1+(−0.642−0.766i)T+(−0.173+0.984i)T2 1 + (-0.642 - 0.766i)T + (-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

−8.435600656820652202935685004320, −7.83531513714842721794740981663, −7.11790872596884508231022845742, −6.29271175231519682124500594598, −5.64013421605171804396830491855, −5.20965850207606028506237884281, −4.65712669219002156218858512658, −3.28086175450440497017471336957, −1.69968242447638502770920694400, −1.22646178390200968141256250067, 1.75970151453805307822597708042, 2.33094083713942036630560914682, 3.52009903605935007566659985082, 4.43488025879247029141618963240, 4.69839876431917125472767295960, 5.66875757650220144508670130383, 6.56535952342299081619968610698, 7.21396120487063478747909778135, 8.260190637850152282745270959893, 9.138111742761920786244593687545

Graph of the ZZ-function along the critical line