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Properties

Label 2-1815-165.128-c0-0-1
Degree $2$
Conductor $1815$
Sign $0.995 - 0.0983i$
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 + (−0.951 + 0.309i)4-s + (−0.587 − 0.809i)5-s + (−0.809 − 0.587i)9-s − 0.999i·12-s + (0.951 − 0.309i)15-s + (0.809 − 0.587i)16-s + (0.809 + 0.587i)20-s + (1 − i)23-s + (−0.309 + 0.951i)25-s + (0.809 − 0.587i)27-s + (0.951 + 0.309i)36-s + (1.26 + 0.642i)37-s + 0.999i·45-s + (1.26 − 0.642i)47-s + (0.309 + 0.951i)48-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s + (−0.951 + 0.309i)4-s + (−0.587 − 0.809i)5-s + (−0.809 − 0.587i)9-s − 0.999i·12-s + (0.951 − 0.309i)15-s + (0.809 − 0.587i)16-s + (0.809 + 0.587i)20-s + (1 − i)23-s + (−0.309 + 0.951i)25-s + (0.809 − 0.587i)27-s + (0.951 + 0.309i)36-s + (1.26 + 0.642i)37-s + 0.999i·45-s + (1.26 − 0.642i)47-s + (0.309 + 0.951i)48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.995 - 0.0983i$
Analytic conductor: \(0.905802\)
Root analytic conductor: \(0.951736\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :0),\ 0.995 - 0.0983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6629787355\)
\(L(\frac12)\) \(\approx\) \(0.6629787355\)
\(L(1)\) 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 \( 1 + (0.587 + 0.809i)T \)
11 \( 1 \)
good2 \( 1 + (0.951 - 0.309i)T^{2} \)
7 \( 1 + (-0.587 - 0.809i)T^{2} \)
13 \( 1 + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (-0.951 - 0.309i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (-1 + i)T - iT^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-1.26 - 0.642i)T + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (-1.26 + 0.642i)T + (0.587 - 0.809i)T^{2} \)
53 \( 1 + (-1.39 + 0.221i)T + (0.951 - 0.309i)T^{2} \)
59 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 + 0.951i)T^{2} \)
67 \( 1 + (1 - i)T - iT^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.587 + 0.809i)T^{2} \)
79 \( 1 + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 + 0.309i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-1.39 + 0.221i)T + (0.951 - 0.309i)T^{2} \)
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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

−9.329354195025545270868385143207, −8.775085806514129259151574810176, −8.234880167597323679026659690122, −7.25087167302688441506300371756, −5.97918207199958903897698257371, −5.12301698253998268913343042641, −4.51618035672656862583713452821, −3.91596923568782009851565134911, −2.92115350341860499658582044663, −0.73530033007437344671679204120, 0.984108919935941527963002700901, 2.47906779845302668674434001793, 3.55232743575295611350509124108, 4.54514131807631793787508120002, 5.57334711661914903807467398942, 6.19854981359383649129439592577, 7.29006788726664174758784312320, 7.63551316817235330080576390775, 8.642059419176399851350935101021, 9.282256433575643642399794491816

Graph of the $Z$-function along the critical line