Properties

Label 2-1815-1.1-c1-0-30
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  − 1.09·2-s + 3-s − 0.800·4-s + 5-s − 1.09·6-s + 0.705·7-s + 3.06·8-s + 9-s − 1.09·10-s − 0.800·12-s + 4.71·13-s − 0.772·14-s + 15-s − 1.75·16-s + 7.78·17-s − 1.09·18-s + 1.19·19-s − 0.800·20-s + 0.705·21-s − 6.89·23-s + 3.06·24-s + 25-s − 5.16·26-s + 27-s − 0.564·28-s − 1.32·29-s − 1.09·30-s + ⋯
L(s)  = 1  − 0.774·2-s + 0.577·3-s − 0.400·4-s + 0.447·5-s − 0.447·6-s + 0.266·7-s + 1.08·8-s + 0.333·9-s − 0.346·10-s − 0.231·12-s + 1.30·13-s − 0.206·14-s + 0.258·15-s − 0.439·16-s + 1.88·17-s − 0.258·18-s + 0.275·19-s − 0.178·20-s + 0.153·21-s − 1.43·23-s + 0.626·24-s + 0.200·25-s − 1.01·26-s + 0.192·27-s − 0.106·28-s − 0.246·29-s − 0.199·30-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

\(L(1)\) \(\approx\) \(1.581304342\)
\(L(\frac12)\) \(\approx\) \(1.581304342\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.09T + 2T^{2} \)
7 \( 1 - 0.705T + 7T^{2} \)
13 \( 1 - 4.71T + 13T^{2} \)
17 \( 1 - 7.78T + 17T^{2} \)
19 \( 1 - 1.19T + 19T^{2} \)
23 \( 1 + 6.89T + 23T^{2} \)
29 \( 1 + 1.32T + 29T^{2} \)
31 \( 1 + 7.68T + 31T^{2} \)
37 \( 1 - 8.43T + 37T^{2} \)
41 \( 1 - 0.232T + 41T^{2} \)
43 \( 1 + 7.32T + 43T^{2} \)
47 \( 1 - 8.32T + 47T^{2} \)
53 \( 1 - 6.82T + 53T^{2} \)
59 \( 1 + 3.54T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 + 2.04T + 67T^{2} \)
71 \( 1 - 0.670T + 71T^{2} \)
73 \( 1 - 5.00T + 73T^{2} \)
79 \( 1 + 2.28T + 79T^{2} \)
83 \( 1 - 2.10T + 83T^{2} \)
89 \( 1 + 3.34T + 89T^{2} \)
97 \( 1 - 3.32T + 97T^{2} \)
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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

−9.309689683135686303693937874371, −8.428316462031846871323300429630, −7.985083877881487304431373588212, −7.26000225419719034099942196086, −5.99746669749041330780370836729, −5.32557192309145897798087092838, −4.09765111183345372406199364681, −3.40930732280279547373074580312, −1.92545257066914544775461073227, −1.02978008896587462937853714286, 1.02978008896587462937853714286, 1.92545257066914544775461073227, 3.40930732280279547373074580312, 4.09765111183345372406199364681, 5.32557192309145897798087092838, 5.99746669749041330780370836729, 7.26000225419719034099942196086, 7.985083877881487304431373588212, 8.428316462031846871323300429630, 9.309689683135686303693937874371

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