Properties

Label 2-1815-1.1-c1-0-52
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  − 2.77·2-s + 3-s + 5.68·4-s − 5-s − 2.77·6-s + 2.27·7-s − 10.2·8-s + 9-s + 2.77·10-s + 5.68·12-s + 0.435·13-s − 6.31·14-s − 15-s + 16.9·16-s − 5·17-s − 2.77·18-s − 4.69·19-s − 5.68·20-s + 2.27·21-s − 0.845·23-s − 10.2·24-s + 25-s − 1.20·26-s + 27-s + 12.9·28-s + 2.65·29-s + 2.77·30-s + ⋯
L(s)  = 1  − 1.96·2-s + 0.577·3-s + 2.84·4-s − 0.447·5-s − 1.13·6-s + 0.860·7-s − 3.61·8-s + 0.333·9-s + 0.876·10-s + 1.64·12-s + 0.120·13-s − 1.68·14-s − 0.258·15-s + 4.23·16-s − 1.21·17-s − 0.653·18-s − 1.07·19-s − 1.27·20-s + 0.497·21-s − 0.176·23-s − 2.08·24-s + 0.200·25-s − 0.236·26-s + 0.192·27-s + 2.44·28-s + 0.493·29-s + 0.506·30-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 \( 1 + 2.77T + 2T^{2} \)
7 \( 1 - 2.27T + 7T^{2} \)
13 \( 1 - 0.435T + 13T^{2} \)
17 \( 1 + 5T + 17T^{2} \)
19 \( 1 + 4.69T + 19T^{2} \)
23 \( 1 + 0.845T + 23T^{2} \)
29 \( 1 - 2.65T + 29T^{2} \)
31 \( 1 + 4.66T + 31T^{2} \)
37 \( 1 + 8.86T + 37T^{2} \)
41 \( 1 - 4.29T + 41T^{2} \)
43 \( 1 + 7.00T + 43T^{2} \)
47 \( 1 - 0.468T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 3.93T + 59T^{2} \)
61 \( 1 + 2.96T + 61T^{2} \)
67 \( 1 + 2.47T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 8.42T + 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 - 5.92T + 83T^{2} \)
89 \( 1 - 5.89T + 89T^{2} \)
97 \( 1 + 8.64T + 97T^{2} \)
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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

−8.697260307898938623119553521982, −8.364441925146093269911588984590, −7.60779189721802670805055577942, −6.91887833325000422313581293837, −6.15285424256003819915166534781, −4.70991941834371149458067458686, −3.43124774553102377367010622118, −2.27148852722030168061340599190, −1.56809857844878514586786121706, 0, 1.56809857844878514586786121706, 2.27148852722030168061340599190, 3.43124774553102377367010622118, 4.70991941834371149458067458686, 6.15285424256003819915166534781, 6.91887833325000422313581293837, 7.60779189721802670805055577942, 8.364441925146093269911588984590, 8.697260307898938623119553521982

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