Properties

Label 2-1215-15.14-c0-0-3
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  + 0.347·2-s − 0.879·4-s + 5-s − 0.652·8-s + 0.347·10-s + 0.652·16-s + 1.53·17-s + 1.53·19-s − 0.879·20-s − 1.87·23-s + 25-s + 0.347·31-s + 0.879·32-s + 0.532·34-s + 0.532·38-s − 0.652·40-s − 0.652·46-s − 47-s + 49-s + 0.347·50-s − 1.87·53-s + 0.347·61-s + 0.120·62-s − 0.347·64-s − 1.34·68-s − 1.34·76-s − 1.87·79-s + ⋯
L(s)  = 1  + 0.347·2-s − 0.879·4-s + 5-s − 0.652·8-s + 0.347·10-s + 0.652·16-s + 1.53·17-s + 1.53·19-s − 0.879·20-s − 1.87·23-s + 25-s + 0.347·31-s + 0.879·32-s + 0.532·34-s + 0.532·38-s − 0.652·40-s − 0.652·46-s − 47-s + 49-s + 0.347·50-s − 1.87·53-s + 0.347·61-s + 0.120·62-s − 0.347·64-s − 1.34·68-s − 1.34·76-s − 1.87·79-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

\(L(\frac{1}{2})\) \(\approx\) \(1.237513693\)
\(L(\frac12)\) \(\approx\) \(1.237513693\)
\(L(1)\) 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 \( 1 - 0.347T + T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 1.53T + T^{2} \)
19 \( 1 - 1.53T + T^{2} \)
23 \( 1 + 1.87T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - 0.347T + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 + 1.87T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 0.347T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 1.87T + T^{2} \)
83 \( 1 - 1.53T + T^{2} \)
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

−9.781307757433529506760367936838, −9.405839217324471597078496058886, −8.298220456246289278574709478254, −7.61319724634937674959421002224, −6.29199373399859431863493072110, −5.60670509948447307850626711930, −4.99674377321321699744893512959, −3.81434641916160081822569221868, −2.89755232741779050442988873666, −1.36087344442962051706943086223, 1.36087344442962051706943086223, 2.89755232741779050442988873666, 3.81434641916160081822569221868, 4.99674377321321699744893512959, 5.60670509948447307850626711930, 6.29199373399859431863493072110, 7.61319724634937674959421002224, 8.298220456246289278574709478254, 9.405839217324471597078496058886, 9.781307757433529506760367936838

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