Properties

Label 2-1815-1.1-c1-0-53
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.35·2-s + 3-s + 3.54·4-s + 5-s + 2.35·6-s − 0.193·7-s + 3.64·8-s + 9-s + 2.35·10-s + 3.54·12-s − 0.973·13-s − 0.455·14-s + 15-s + 1.49·16-s + 2.67·17-s + 2.35·18-s + 5.54·19-s + 3.54·20-s − 0.193·21-s − 4.80·23-s + 3.64·24-s + 25-s − 2.29·26-s + 27-s − 0.686·28-s + 10.1·29-s + 2.35·30-s + ⋯
L(s)  = 1  + 1.66·2-s + 0.577·3-s + 1.77·4-s + 0.447·5-s + 0.961·6-s − 0.0731·7-s + 1.29·8-s + 0.333·9-s + 0.744·10-s + 1.02·12-s − 0.270·13-s − 0.121·14-s + 0.258·15-s + 0.374·16-s + 0.648·17-s + 0.555·18-s + 1.27·19-s + 0.793·20-s − 0.0422·21-s − 1.00·23-s + 0.744·24-s + 0.200·25-s − 0.449·26-s + 0.192·27-s − 0.129·28-s + 1.87·29-s + 0.430·30-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: \(0\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.043062334\)
\(L(\frac12)\) \(\approx\) \(6.043062334\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.35T + 2T^{2} \)
7 \( 1 + 0.193T + 7T^{2} \)
13 \( 1 + 0.973T + 13T^{2} \)
17 \( 1 - 2.67T + 17T^{2} \)
19 \( 1 - 5.54T + 19T^{2} \)
23 \( 1 + 4.80T + 23T^{2} \)
29 \( 1 - 10.1T + 29T^{2} \)
31 \( 1 + 2.50T + 31T^{2} \)
37 \( 1 - 5.71T + 37T^{2} \)
41 \( 1 - 8.27T + 41T^{2} \)
43 \( 1 + 5.11T + 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 + 9.28T + 53T^{2} \)
59 \( 1 + 11.2T + 59T^{2} \)
61 \( 1 + 1.20T + 61T^{2} \)
67 \( 1 - 3.25T + 67T^{2} \)
71 \( 1 - 5.99T + 71T^{2} \)
73 \( 1 + 3.30T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 8.31T + 83T^{2} \)
89 \( 1 - 7.34T + 89T^{2} \)
97 \( 1 + 15.8T + 97T^{2} \)
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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

−9.510405745265142590518942471297, −8.254137922185241416476797677281, −7.53359435945286096346273318145, −6.57535766079166193415333779326, −5.93871752012208241956761582382, −5.04677735323847615249620167287, −4.37624537970547017974853502395, −3.28438806573780059091644503585, −2.77833282591387082970962317041, −1.57109557227935181812235434246, 1.57109557227935181812235434246, 2.77833282591387082970962317041, 3.28438806573780059091644503585, 4.37624537970547017974853502395, 5.04677735323847615249620167287, 5.93871752012208241956761582382, 6.57535766079166193415333779326, 7.53359435945286096346273318145, 8.254137922185241416476797677281, 9.510405745265142590518942471297

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