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Properties

Label 2-1815-165.59-c0-0-6
Degree 22
Conductor 18151815
Sign −0.642+0.766i-0.642 + 0.766i
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  + (−0.309 + 0.951i)3-s + (−0.309 − 0.951i)4-s + (−0.809 + 0.587i)5-s + (−0.809 − 0.587i)9-s + 0.999·12-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.809 + 0.587i)20-s − 2·23-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + (−1.61 − 1.17i)31-s + (−0.309 + 0.951i)36-s + 0.999·45-s + (0.618 − 1.90i)47-s + (−0.309 − 0.951i)48-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s + (−0.309 − 0.951i)4-s + (−0.809 + 0.587i)5-s + (−0.809 − 0.587i)9-s + 0.999·12-s + (−0.309 − 0.951i)15-s + (−0.809 + 0.587i)16-s + (0.809 + 0.587i)20-s − 2·23-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + (−1.61 − 1.17i)31-s + (−0.309 + 0.951i)36-s + 0.999·45-s + (0.618 − 1.90i)47-s + (−0.309 − 0.951i)48-s + ⋯

Functional equation

Λ(s)=(1815s/2 ΓC(s) L(s)=((−0.642+0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(1815s/2 ΓC(s) L(s)=((−0.642+0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 18151815    =    3⋅5⋅1123 \cdot 5 \cdot 11^{2}
Sign: −0.642+0.766i-0.642 + 0.766i
Analytic conductor: 0.9058020.905802
Root analytic conductor: 0.9517360.951736
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ1815(1049,⋅)\chi_{1815} (1049, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1815, ( :0), −0.642+0.766i)(2,\ 1815,\ (\ :0),\ -0.642 + 0.766i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.15386045580.1538604558
L(12)L(\frac12) ≈\approx 0.15386045580.1538604558
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+(0.309−0.951i)T 1 + (0.309 - 0.951i)T
5 1+(0.809−0.587i)T 1 + (0.809 - 0.587i)T
11 1 1
good2 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
7 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
13 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)T^{2}
17 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
19 1+(−0.809−0.587i)T2 1 + (-0.809 - 0.587i)T^{2}
23 1+2T+T2 1 + 2T + T^{2}
29 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
31 1+(1.61+1.17i)T+(0.309+0.951i)T2 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2}
37 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
41 1+(0.809+0.587i)T2 1 + (0.809 + 0.587i)T^{2}
43 1−T2 1 - T^{2}
47 1+(−0.618+1.90i)T+(−0.809−0.587i)T2 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2}
53 1+(1.61+1.17i)T+(0.309+0.951i)T2 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2}
59 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
61 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
67 1−T2 1 - T^{2}
71 1+(−0.309+0.951i)T2 1 + (-0.309 + 0.951i)T^{2}
73 1+(0.809−0.587i)T2 1 + (0.809 - 0.587i)T^{2}
79 1+(0.309+0.951i)T2 1 + (0.309 + 0.951i)T^{2}
83 1+(0.309−0.951i)T2 1 + (0.309 - 0.951i)T^{2}
89 1−T2 1 - T^{2}
97 1+(−0.309−0.951i)T2 1 + (-0.309 - 0.951i)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

−9.398807897831135551076206588091, −8.504560777955781884350327921667, −7.67745148583229730935224138478, −6.54470293394561449733858120162, −5.90071889215554155210256010090, −5.06834472441158219404922787744, −4.13870364400120784210035557835, −3.58479043149398759032717981350, −2.14085894493375642298972644180, −0.11657590058233451959682657159, 1.64957419471006557370417787151, 2.97717148550644085183360690348, 3.92738743065864183146225175975, 4.78428214810834825470227494186, 5.75905844211031755709008676512, 6.76223078182398118698394662105, 7.65049717121541997325555262822, 7.943795214674795353093899515017, 8.724250555028151935912127266186, 9.436996876924458651129433406559

Graph of the ZZ-function along the critical line