Properties

Label 2-1215-15.14-c0-0-4
Degree $2$
Conductor $1215$
Sign $1$
Analytic cond. $0.606363$
Root an. cond. $0.778693$
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  + 1.87·2-s + 2.53·4-s − 5-s + 2.87·8-s − 1.87·10-s + 2.87·16-s − 0.347·17-s + 0.347·19-s − 2.53·20-s − 1.53·23-s + 25-s − 1.87·31-s + 2.53·32-s − 0.652·34-s + 0.652·38-s − 2.87·40-s − 2.87·46-s + 47-s + 49-s + 1.87·50-s − 1.53·53-s − 1.87·61-s − 3.53·62-s + 1.87·64-s − 0.879·68-s + 0.879·76-s + 1.53·79-s + ⋯
L(s)  = 1  + 1.87·2-s + 2.53·4-s − 5-s + 2.87·8-s − 1.87·10-s + 2.87·16-s − 0.347·17-s + 0.347·19-s − 2.53·20-s − 1.53·23-s + 25-s − 1.87·31-s + 2.53·32-s − 0.652·34-s + 0.652·38-s − 2.87·40-s − 2.87·46-s + 47-s + 49-s + 1.87·50-s − 1.53·53-s − 1.87·61-s − 3.53·62-s + 1.87·64-s − 0.879·68-s + 0.879·76-s + 1.53·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1215\)    =    \(3^{5} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.606363\)
Root analytic conductor: \(0.778693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1215} (1214, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1215,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.696368910\)
\(L(\frac12)\) \(\approx\) \(2.696368910\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + T \)
good2 \( 1 - 1.87T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 0.347T + T^{2} \)
19 \( 1 - 0.347T + T^{2} \)
23 \( 1 + 1.53T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.87T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T + T^{2} \)
53 \( 1 + 1.53T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.87T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.53T + T^{2} \)
83 \( 1 + 0.347T + T^{2} \)
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

−10.39062278455287463360303570307, −9.053029581490406539810426006191, −7.81414878883878292461954496392, −7.33006665411073151930560875668, −6.36149557355958041185135603413, −5.57149517508684153995894669899, −4.62422241423623700054177698200, −3.92297665967567896912010639822, −3.19662269227392008259292455690, −1.98247246182906572586659465493, 1.98247246182906572586659465493, 3.19662269227392008259292455690, 3.92297665967567896912010639822, 4.62422241423623700054177698200, 5.57149517508684153995894669899, 6.36149557355958041185135603413, 7.33006665411073151930560875668, 7.81414878883878292461954496392, 9.053029581490406539810426006191, 10.39062278455287463360303570307

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