Properties

Label 2-1815-1.1-c1-0-0
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  − 2.13·2-s − 3-s + 2.55·4-s + 5-s + 2.13·6-s − 4.82·7-s − 1.17·8-s + 9-s − 2.13·10-s − 2.55·12-s − 4.26·13-s + 10.2·14-s − 15-s − 2.59·16-s − 4.64·17-s − 2.13·18-s + 6.37·19-s + 2.55·20-s + 4.82·21-s − 5.14·23-s + 1.17·24-s + 25-s + 9.10·26-s − 27-s − 12.3·28-s − 4.26·29-s + 2.13·30-s + ⋯
L(s)  = 1  − 1.50·2-s − 0.577·3-s + 1.27·4-s + 0.447·5-s + 0.871·6-s − 1.82·7-s − 0.416·8-s + 0.333·9-s − 0.674·10-s − 0.736·12-s − 1.18·13-s + 2.74·14-s − 0.258·15-s − 0.647·16-s − 1.12·17-s − 0.502·18-s + 1.46·19-s + 0.570·20-s + 1.05·21-s − 1.07·23-s + 0.240·24-s + 0.200·25-s + 1.78·26-s − 0.192·27-s − 2.32·28-s − 0.792·29-s + 0.389·30-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

\(L(1)\) \(\approx\) \(0.2567248814\)
\(L(\frac12)\) \(\approx\) \(0.2567248814\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.13T + 2T^{2} \)
7 \( 1 + 4.82T + 7T^{2} \)
13 \( 1 + 4.26T + 13T^{2} \)
17 \( 1 + 4.64T + 17T^{2} \)
19 \( 1 - 6.37T + 19T^{2} \)
23 \( 1 + 5.14T + 23T^{2} \)
29 \( 1 + 4.26T + 29T^{2} \)
31 \( 1 + 6.39T + 31T^{2} \)
37 \( 1 - 6.14T + 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 - 1.55T + 43T^{2} \)
47 \( 1 - 5.35T + 47T^{2} \)
53 \( 1 - 2.24T + 53T^{2} \)
59 \( 1 + 13.5T + 59T^{2} \)
61 \( 1 - 6.80T + 61T^{2} \)
67 \( 1 - 6.14T + 67T^{2} \)
71 \( 1 + 10.4T + 71T^{2} \)
73 \( 1 + 5.62T + 73T^{2} \)
79 \( 1 + 16.3T + 79T^{2} \)
83 \( 1 + 6.17T + 83T^{2} \)
89 \( 1 + 1.39T + 89T^{2} \)
97 \( 1 - 3.93T + 97T^{2} \)
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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

−9.433521383817510391256587153205, −8.846609137886775650501700921811, −7.46930989901366587398848836934, −7.17707041498169175255894352541, −6.28344102183857247889479581418, −5.57381021842940058164746225460, −4.29088769115292869786994751798, −2.97967487707175733839600524765, −1.96169091183883981829829126970, −0.42237175358582420595043809135, 0.42237175358582420595043809135, 1.96169091183883981829829126970, 2.97967487707175733839600524765, 4.29088769115292869786994751798, 5.57381021842940058164746225460, 6.28344102183857247889479581418, 7.17707041498169175255894352541, 7.46930989901366587398848836934, 8.846609137886775650501700921811, 9.433521383817510391256587153205

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