Properties

Label 2-1215-45.14-c0-0-2
Degree $2$
Conductor $1215$
Sign $0.766 + 0.642i$
Analytic cond. $0.606363$
Root an. cond. $0.778693$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 − 0.300i)2-s + (0.439 − 0.761i)4-s + (−0.5 + 0.866i)5-s − 0.652·8-s + 0.347·10-s + (−0.326 − 0.565i)16-s + 1.53·17-s + 1.53·19-s + (0.439 + 0.761i)20-s + (0.939 − 1.62i)23-s + (−0.499 − 0.866i)25-s + (−0.173 + 0.300i)31-s + (−0.439 + 0.761i)32-s + (−0.266 − 0.460i)34-s + (−0.266 − 0.460i)38-s + ⋯
L(s)  = 1  + (−0.173 − 0.300i)2-s + (0.439 − 0.761i)4-s + (−0.5 + 0.866i)5-s − 0.652·8-s + 0.347·10-s + (−0.326 − 0.565i)16-s + 1.53·17-s + 1.53·19-s + (0.439 + 0.761i)20-s + (0.939 − 1.62i)23-s + (−0.499 − 0.866i)25-s + (−0.173 + 0.300i)31-s + (−0.439 + 0.761i)32-s + (−0.266 − 0.460i)34-s + (−0.266 − 0.460i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1215\)    =    \(3^{5} \cdot 5\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(0.606363\)
Root analytic conductor: \(0.778693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1215} (404, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1215,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.018200811\)
\(L(\frac12)\) \(\approx\) \(1.018200811\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 - 1.53T + T^{2} \)
19 \( 1 - 1.53T + T^{2} \)
23 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + 1.87T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.920430596999991108737085537544, −9.324453445881569782770703487801, −8.093593958384109684352536396740, −7.33599898328259186261373855491, −6.56558411674505011492020728747, −5.72844551324108532869195374195, −4.77327750538101906887754852075, −3.32463952586796080624503708210, −2.72102587627716572398435657778, −1.17277772312568379790859808718, 1.38852610148519858640062378496, 3.14026499578317593771942932448, 3.70257827248639002704683079580, 5.08042239849510953222938732151, 5.71656669853675319095296096271, 7.04039835569254357557949484162, 7.67432023701203862085404388204, 8.147981742945190504268267422617, 9.216843577078741185593390354310, 9.670912403612827139073759912531

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