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Properties

Label 2-1815-165.119-c0-0-5
Degree 22
Conductor 18151815
Sign 0.957−0.288i0.957 - 0.288i
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  + (0.809 − 0.587i)3-s + (0.809 + 0.587i)4-s + (0.309 + 0.951i)5-s + (0.309 − 0.951i)9-s + 12-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.309 + 0.951i)20-s − 2·23-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (0.618 − 1.90i)31-s + (0.809 − 0.587i)36-s + 45-s + (−1.61 + 1.17i)47-s + (0.809 + 0.587i)48-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (0.809 + 0.587i)4-s + (0.309 + 0.951i)5-s + (0.309 − 0.951i)9-s + 12-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.309 + 0.951i)20-s − 2·23-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (0.618 − 1.90i)31-s + (0.809 − 0.587i)36-s + 45-s + (−1.61 + 1.17i)47-s + (0.809 + 0.587i)48-s + ⋯

Functional equation

Λ(s)=(1815s/2 ΓC(s) L(s)=((0.957−0.288i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1815s/2 ΓC(s) L(s)=((0.957−0.288i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18151815    =    3⋅5⋅1123 \cdot 5 \cdot 11^{2}
Sign: 0.957−0.288i0.957 - 0.288i
Analytic conductor: 0.9058020.905802
Root analytic conductor: 0.9517360.951736
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1815(614,⋅)\chi_{1815} (614, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1815, ( :0), 0.957−0.288i)(2,\ 1815,\ (\ :0),\ 0.957 - 0.288i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 1.8406627641.840662764
L(12)L(\frac12) ≈\approx 1.8406627641.840662764
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+(−0.809−0.587i)T2 1 + (-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+(−0.809+0.587i)T2 1 + (-0.809 + 0.587i)T^{2}
19 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
23 1+2T+T2 1 + 2T + T^{2}
29 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
31 1+(−0.618+1.90i)T+(−0.809−0.587i)T2 1 + (-0.618 + 1.90i)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+(1.61−1.17i)T+(0.309−0.951i)T2 1 + (1.61 - 1.17i)T + (0.309 - 0.951i)T^{2}
53 1+(−0.618+1.90i)T+(−0.809−0.587i)T2 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2}
59 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
61 1+(−0.809+0.587i)T2 1 + (-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+(−0.809−0.587i)T2 1 + (-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

−9.639425621504580191521693573095, −8.378314304960135305199617144675, −7.85804492829460603653058399781, −7.25104127957423472371838856686, −6.35556549788546608468668614745, −5.99605176218627479266010625220, −4.15684443637186762001407951090, −3.38144855573687194262805480722, −2.49155590994682791374078139547, −1.86586363846880906882406590842, 1.51730003409549472460613265647, 2.34137705233346790571998974515, 3.49856094264172595686549764941, 4.54857317189007985036241576170, 5.30329429578696702667737345503, 6.11449448705083067891280772583, 7.10900269519336955703568354151, 8.073574926583927336260868079150, 8.619669449526333634203592261444, 9.511232506295505346981006049401

Graph of the ZZ-function along the critical line