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Properties

Label 2-2793-399.314-c0-0-1
Degree 22
Conductor 27932793
Sign 0.349+0.937i0.349 + 0.937i
Analytic cond. 1.393881.39388
Root an. cond. 1.180631.18063
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)3-s + (0.939 − 0.342i)4-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)12-s + (−0.266 − 0.223i)13-s + (0.766 − 0.642i)16-s + (−0.5 − 0.866i)19-s + (−0.766 − 0.642i)25-s + (−0.500 − 0.866i)27-s + (−0.939 + 1.62i)31-s + (−0.173 − 0.984i)36-s + 1.28i·37-s − 0.347·39-s + (1.43 + 0.524i)43-s + (0.173 − 0.984i)48-s + ⋯
L(s)  = 1  + (0.766 − 0.642i)3-s + (0.939 − 0.342i)4-s + (0.173 − 0.984i)9-s + (0.5 − 0.866i)12-s + (−0.266 − 0.223i)13-s + (0.766 − 0.642i)16-s + (−0.5 − 0.866i)19-s + (−0.766 − 0.642i)25-s + (−0.500 − 0.866i)27-s + (−0.939 + 1.62i)31-s + (−0.173 − 0.984i)36-s + 1.28i·37-s − 0.347·39-s + (1.43 + 0.524i)43-s + (0.173 − 0.984i)48-s + ⋯

Functional equation

Λ(s)=(2793s/2 ΓC(s) L(s)=((0.349+0.937i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2793s/2 ΓC(s) L(s)=((0.349+0.937i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 27932793    =    3⋅72⋅193 \cdot 7^{2} \cdot 19
Sign: 0.349+0.937i0.349 + 0.937i
Analytic conductor: 1.393881.39388
Root analytic conductor: 1.180631.18063
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2793(1910,⋅)\chi_{2793} (1910, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 2793, ( :0), 0.349+0.937i)(2,\ 2793,\ (\ :0),\ 0.349 + 0.937i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.9371423521.937142352
L(12)L(\frac12) ≈\approx 1.9371423521.937142352
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+(−0.766+0.642i)T 1 + (-0.766 + 0.642i)T
7 1 1
19 1+(0.5+0.866i)T 1 + (0.5 + 0.866i)T
good2 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
5 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
11 1+(0.5−0.866i)T2 1 + (0.5 - 0.866i)T^{2}
13 1+(0.266+0.223i)T+(0.173+0.984i)T2 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2}
17 1+(−0.939+0.342i)T2 1 + (-0.939 + 0.342i)T^{2}
23 1+(−0.766+0.642i)T2 1 + (-0.766 + 0.642i)T^{2}
29 1+(−0.939−0.342i)T2 1 + (-0.939 - 0.342i)T^{2}
31 1+(0.939−1.62i)T+(−0.5−0.866i)T2 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2}
37 1−1.28iT−T2 1 - 1.28iT - T^{2}
41 1+(−0.173+0.984i)T2 1 + (-0.173 + 0.984i)T^{2}
43 1+(−1.43−0.524i)T+(0.766+0.642i)T2 1 + (-1.43 - 0.524i)T + (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+(0.939−0.342i)T2 1 + (0.939 - 0.342i)T^{2}
61 1+(−0.233−0.642i)T+(−0.766+0.642i)T2 1 + (-0.233 - 0.642i)T + (-0.766 + 0.642i)T^{2}
67 1+(−1.93−0.342i)T+(0.939+0.342i)T2 1 + (-1.93 - 0.342i)T + (0.939 + 0.342i)T^{2}
71 1+(0.766+0.642i)T2 1 + (0.766 + 0.642i)T^{2}
73 1+(−1.26−1.50i)T+(−0.173+0.984i)T2 1 + (-1.26 - 1.50i)T + (-0.173 + 0.984i)T^{2}
79 1+(0.439+0.524i)T+(−0.173+0.984i)T2 1 + (0.439 + 0.524i)T + (-0.173 + 0.984i)T^{2}
83 1+(−0.5−0.866i)T2 1 + (-0.5 - 0.866i)T^{2}
89 1+(−0.173−0.984i)T2 1 + (-0.173 - 0.984i)T^{2}
97 1+(0.173+0.984i)T+(−0.939+0.342i)T2 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)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

−8.699562342697277958169620877922, −8.039734006817836410426993303086, −7.23084243871308853287342651772, −6.74142786026584923520610596649, −6.00917347821099060174841011530, −5.04716184733886218186966558161, −3.85947627675494708454267819506, −2.86113413774530266778410411599, −2.23856253907122666516986393406, −1.16762126995952942506135026883, 1.91362060488317683748138259465, 2.44926523662053302857148600610, 3.68569873832408396817612834463, 4.00147513394692330703234632209, 5.32942311006452604177710729938, 6.03451959409928460416989555727, 7.07933659208487522002044404643, 7.74545341111830075819974387600, 8.204290215789956263068567153476, 9.284587045322667655575964261810

Graph of the ZZ-function along the critical line