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Properties

Label 2-1815-165.59-c0-0-6
Degree $2$
Conductor $1815$
Sign $-0.642 + 0.766i$
Analytic cond. $0.905802$
Root an. cond. $0.951736$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.642 + 0.766i$
Analytic conductor: \(0.905802\)
Root analytic conductor: \(0.951736\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :0),\ -0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1538604558\)
\(L(\frac12)\) \(\approx\) \(0.1538604558\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + 2T + T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 - 0.587i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.309 - 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.309 - 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

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 $Z$-function along the critical line