Properties

Label 2-1815-15.14-c0-0-9
Degree $2$
Conductor $1815$
Sign $1$
Analytic cond. $0.905802$
Root an. cond. $0.951736$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73·2-s + 3-s + 1.99·4-s − 5-s + 1.73·6-s + 1.73·8-s + 9-s − 1.73·10-s + 1.99·12-s − 15-s + 0.999·16-s − 1.73·17-s + 1.73·18-s − 1.99·20-s − 23-s + 1.73·24-s + 25-s + 27-s − 1.73·30-s − 31-s − 2.99·34-s + 1.99·36-s − 1.73·40-s − 45-s − 1.73·46-s + 47-s + 0.999·48-s + ⋯
L(s)  = 1  + 1.73·2-s + 3-s + 1.99·4-s − 5-s + 1.73·6-s + 1.73·8-s + 9-s − 1.73·10-s + 1.99·12-s − 15-s + 0.999·16-s − 1.73·17-s + 1.73·18-s − 1.99·20-s − 23-s + 1.73·24-s + 25-s + 27-s − 1.73·30-s − 31-s − 2.99·34-s + 1.99·36-s − 1.73·40-s − 45-s − 1.73·46-s + 47-s + 0.999·48-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.905802\)
Root analytic conductor: \(0.951736\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1574, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.449569442\)
\(L(\frac12)\) \(\approx\) \(3.449569442\)
\(L(1)\) 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 \( 1 - 1.73T + T^{2} \)
7 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.73T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.73T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.73T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.264180899765024003070754522505, −8.570255334305873910561405012797, −7.61171311947797003367397914238, −7.03435184078224158050080459254, −6.23789872987372127702708394998, −5.07644779908904633346197681159, −4.16578495649104800449671578928, −3.90090377276499587792199817408, −2.84666865869189123637092785498, −2.03033035824921639371144693890, 2.03033035824921639371144693890, 2.84666865869189123637092785498, 3.90090377276499587792199817408, 4.16578495649104800449671578928, 5.07644779908904633346197681159, 6.23789872987372127702708394998, 7.03435184078224158050080459254, 7.61171311947797003367397914238, 8.570255334305873910561405012797, 9.264180899765024003070754522505

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