Properties

Label 2-1215-15.14-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.6020696505\)
\(L(\frac12)\) \(\approx\) \(0.6020696505\)
\(L(1)\) 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

−9.806111112338851928828216428101, −8.848975733966765300053879982991, −8.621696821443403005276224123950, −7.41635556506473830273969777476, −7.03114400708906439862728711072, −5.52228458483151498142393868828, −4.69000145792961307767509751242, −3.94716475438909020550293766369, −2.85880377574486163417854967499, −0.938118664095298493348676384056, 0.938118664095298493348676384056, 2.85880377574486163417854967499, 3.94716475438909020550293766369, 4.69000145792961307767509751242, 5.52228458483151498142393868828, 7.03114400708906439862728711072, 7.41635556506473830273969777476, 8.621696821443403005276224123950, 8.848975733966765300053879982991, 9.806111112338851928828216428101

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