Properties

Label 2-1815-1.1-c1-0-70
Degree $2$
Conductor $1815$
Sign $-1$
Analytic cond. $14.4928$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s + 3-s + 0.999·4-s − 5-s + 1.73·6-s − 2·7-s − 1.73·8-s + 9-s − 1.73·10-s + 0.999·12-s − 5.46·13-s − 3.46·14-s − 15-s − 5·16-s + 1.73·18-s − 5.46·19-s − 0.999·20-s − 2·21-s + 6.92·23-s − 1.73·24-s + 25-s − 9.46·26-s + 27-s − 1.99·28-s + 3.46·29-s − 1.73·30-s − 10.9·31-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.577·3-s + 0.499·4-s − 0.447·5-s + 0.707·6-s − 0.755·7-s − 0.612·8-s + 0.333·9-s − 0.547·10-s + 0.288·12-s − 1.51·13-s − 0.925·14-s − 0.258·15-s − 1.25·16-s + 0.408·18-s − 1.25·19-s − 0.223·20-s − 0.436·21-s + 1.44·23-s − 0.353·24-s + 0.200·25-s − 1.85·26-s + 0.192·27-s − 0.377·28-s + 0.643·29-s − 0.316·30-s − 1.96·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(14.4928\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 + T \)
11 \( 1 \)
good2 \( 1 - 1.73T + 2T^{2} \)
7 \( 1 + 2T + 7T^{2} \)
13 \( 1 + 5.46T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.46T + 19T^{2} \)
23 \( 1 - 6.92T + 23T^{2} \)
29 \( 1 - 3.46T + 29T^{2} \)
31 \( 1 + 10.9T + 31T^{2} \)
37 \( 1 + 4.92T + 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 - 4.92T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 - 0.928T + 53T^{2} \)
59 \( 1 + 6.92T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 13.8T + 71T^{2} \)
73 \( 1 - 8.39T + 73T^{2} \)
79 \( 1 - 6.53T + 79T^{2} \)
83 \( 1 + 8.53T + 83T^{2} \)
89 \( 1 - 0.928T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.012646344264965441331115386482, −8.013413489399351804721201627120, −6.99571487166376361457011261989, −6.57172900199190558672907389573, −5.31363866241500678720553305179, −4.72565192280137514755377836435, −3.78473576132666394055654531978, −3.10289076230485859240868530454, −2.22860000122942472953322067492, 0, 2.22860000122942472953322067492, 3.10289076230485859240868530454, 3.78473576132666394055654531978, 4.72565192280137514755377836435, 5.31363866241500678720553305179, 6.57172900199190558672907389573, 6.99571487166376361457011261989, 8.013413489399351804721201627120, 9.012646344264965441331115386482

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