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Properties

Label 2-1815-165.104-c0-0-3
Degree 22
Conductor 18151815
Sign −0.923−0.382i-0.923 - 0.382i
Analytic cond. 0.9058020.905802
Root an. cond. 0.9517360.951736
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  + (0.535 + 1.64i)2-s + (0.809 + 0.587i)3-s + (−1.61 + 1.17i)4-s + (0.309 − 0.951i)5-s + (−0.535 + 1.64i)6-s + (−1.40 − 1.01i)8-s + (0.309 + 0.951i)9-s + 1.73·10-s − 2·12-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (−0.535 + 1.64i)17-s + (−1.40 + 1.01i)18-s + (0.618 + 1.90i)20-s + 23-s + (−0.535 − 1.64i)24-s + ⋯
L(s)  = 1  + (0.535 + 1.64i)2-s + (0.809 + 0.587i)3-s + (−1.61 + 1.17i)4-s + (0.309 − 0.951i)5-s + (−0.535 + 1.64i)6-s + (−1.40 − 1.01i)8-s + (0.309 + 0.951i)9-s + 1.73·10-s − 2·12-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (−0.535 + 1.64i)17-s + (−1.40 + 1.01i)18-s + (0.618 + 1.90i)20-s + 23-s + (−0.535 − 1.64i)24-s + ⋯

Functional equation

Λ(s)=(1815s/2 ΓC(s) L(s)=((−0.923−0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1815s/2 ΓC(s) L(s)=((−0.923−0.382i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18151815    =    3⋅5⋅1123 \cdot 5 \cdot 11^{2}
Sign: −0.923−0.382i-0.923 - 0.382i
Analytic conductor: 0.9058020.905802
Root analytic conductor: 0.9517360.951736
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1815(269,⋅)\chi_{1815} (269, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1815, ( :0), −0.923−0.382i)(2,\ 1815,\ (\ :0),\ -0.923 - 0.382i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8571186781.857118678
L(12)L(\frac12) ≈\approx 1.8571186781.857118678
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+(−0.809−0.587i)T 1 + (-0.809 - 0.587i)T
5 1+(−0.309+0.951i)T 1 + (-0.309 + 0.951i)T
11 1 1
good2 1+(−0.535−1.64i)T+(−0.809+0.587i)T2 1 + (-0.535 - 1.64i)T + (-0.809 + 0.587i)T^{2}
7 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
13 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
17 1+(0.535−1.64i)T+(−0.809−0.587i)T2 1 + (0.535 - 1.64i)T + (-0.809 - 0.587i)T^{2}
19 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
23 1−T+T2 1 - T + T^{2}
29 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
31 1+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
37 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
41 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
43 1−T2 1 - T^{2}
47 1+(−0.809−0.587i)T+(0.309+0.951i)T2 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2}
53 1+(0.309+0.951i)T+(−0.809+0.587i)T2 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2}
59 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
61 1+(−0.535+1.64i)T+(−0.809−0.587i)T2 1 + (-0.535 + 1.64i)T + (-0.809 - 0.587i)T^{2}
67 1−T2 1 - T^{2}
71 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
73 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
79 1+(0.535+1.64i)T+(−0.809+0.587i)T2 1 + (0.535 + 1.64i)T + (-0.809 + 0.587i)T^{2}
83 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
89 1−T2 1 - T^{2}
97 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)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

−9.314005779849021234843918916898, −8.763734279134567736726143618104, −8.217415218671791964751048329990, −7.55984875421439696565619117876, −6.54208094713616025033022716379, −5.72980743888037656104507117587, −4.97589579020083206326588579799, −4.28685672180684515957588240419, −3.57420621060066205574060693253, −1.97591358580546051465031042557, 1.20873786085515939217120398307, 2.43816103374506605957109761483, 2.82047950132424410147713282942, 3.65741771849464972727118255861, 4.66764074677438599133585514521, 5.71019950575326746720396149637, 6.92771655738706425327503590801, 7.35702753424612828922623798131, 8.747863501192013584990852543842, 9.306930685475737456956931361758

Graph of the ZZ-function along the critical line