Properties

Label 2-1815-1.1-c1-0-18
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  + 1.61·2-s − 3-s + 0.618·4-s − 5-s − 1.61·6-s − 1.23·7-s − 2.23·8-s + 9-s − 1.61·10-s − 0.618·12-s + 0.763·13-s − 2.00·14-s + 15-s − 4.85·16-s + 3.47·17-s + 1.61·18-s − 2.47·19-s − 0.618·20-s + 1.23·21-s + 8.70·23-s + 2.23·24-s + 25-s + 1.23·26-s − 27-s − 0.763·28-s + 3.23·29-s + 1.61·30-s + ⋯
L(s)  = 1  + 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.447·5-s − 0.660·6-s − 0.467·7-s − 0.790·8-s + 0.333·9-s − 0.511·10-s − 0.178·12-s + 0.211·13-s − 0.534·14-s + 0.258·15-s − 1.21·16-s + 0.842·17-s + 0.381·18-s − 0.567·19-s − 0.138·20-s + 0.269·21-s + 1.81·23-s + 0.456·24-s + 0.200·25-s + 0.242·26-s − 0.192·27-s − 0.144·28-s + 0.600·29-s + 0.295·30-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

\(L(1)\) \(\approx\) \(1.944862464\)
\(L(\frac12)\) \(\approx\) \(1.944862464\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.61T + 2T^{2} \)
7 \( 1 + 1.23T + 7T^{2} \)
13 \( 1 - 0.763T + 13T^{2} \)
17 \( 1 - 3.47T + 17T^{2} \)
19 \( 1 + 2.47T + 19T^{2} \)
23 \( 1 - 8.70T + 23T^{2} \)
29 \( 1 - 3.23T + 29T^{2} \)
31 \( 1 - 6.70T + 31T^{2} \)
37 \( 1 - 5.23T + 37T^{2} \)
41 \( 1 - 12.4T + 41T^{2} \)
43 \( 1 + 0.763T + 43T^{2} \)
47 \( 1 + 4.70T + 47T^{2} \)
53 \( 1 + 7.94T + 53T^{2} \)
59 \( 1 + 1.70T + 59T^{2} \)
61 \( 1 - 7.47T + 61T^{2} \)
67 \( 1 - 6.76T + 67T^{2} \)
71 \( 1 + 5.52T + 71T^{2} \)
73 \( 1 + 1.52T + 73T^{2} \)
79 \( 1 + 0.708T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 - 9.23T + 89T^{2} \)
97 \( 1 - 17.2T + 97T^{2} \)
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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

−9.325012064678932692986700752254, −8.460909499670961194537051937551, −7.47676152222131388133449886600, −6.50230302422962949453235552369, −6.04276253560562332359592948124, −5.00708051626643676653931728210, −4.48527348226146045173814460850, −3.48742788429213568835127407996, −2.74253557029201212677306955913, −0.830149055478123674746407958472, 0.830149055478123674746407958472, 2.74253557029201212677306955913, 3.48742788429213568835127407996, 4.48527348226146045173814460850, 5.00708051626643676653931728210, 6.04276253560562332359592948124, 6.50230302422962949453235552369, 7.47676152222131388133449886600, 8.460909499670961194537051937551, 9.325012064678932692986700752254

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