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Properties

Label 2-1215-45.29-c0-0-0
Degree $2$
Conductor $1215$
Sign $-0.939 - 0.342i$
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  + (−0.939 + 1.62i)2-s + (−1.26 − 2.19i)4-s + (0.5 + 0.866i)5-s + 2.87·8-s − 1.87·10-s + (−1.43 + 2.49i)16-s − 0.347·17-s + 0.347·19-s + (1.26 − 2.19i)20-s + (0.766 + 1.32i)23-s + (−0.499 + 0.866i)25-s + (0.939 + 1.62i)31-s + (−1.26 − 2.19i)32-s + (0.326 − 0.565i)34-s + (−0.326 + 0.565i)38-s + ⋯
L(s)  = 1  + (−0.939 + 1.62i)2-s + (−1.26 − 2.19i)4-s + (0.5 + 0.866i)5-s + 2.87·8-s − 1.87·10-s + (−1.43 + 2.49i)16-s − 0.347·17-s + 0.347·19-s + (1.26 − 2.19i)20-s + (0.766 + 1.32i)23-s + (−0.499 + 0.866i)25-s + (0.939 + 1.62i)31-s + (−1.26 − 2.19i)32-s + (0.326 − 0.565i)34-s + (−0.326 + 0.565i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1215\)    =    \(3^{5} \cdot 5\)
Sign: $-0.939 - 0.342i$
Analytic conductor: \(0.606363\)
Root analytic conductor: \(0.778693\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1215} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1215,\ (\ :0),\ -0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6347876740\)
\(L(\frac12)\) \(\approx\) \(0.6347876740\)
\(L(1)\) 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 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 0.347T + T^{2} \)
19 \( 1 - 0.347T + T^{2} \)
23 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + 1.53T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
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

−9.863008625724973196852865620271, −9.448720512671904316600894556204, −8.527593835820825129944882131500, −7.74149437841710070197134530633, −6.93392549603838549551660014616, −6.48323753398819499809014771162, −5.54996344895314981816028837252, −4.82455115323385194506942301479, −3.22863440300120722587320186993, −1.53750201066889805159593740776, 0.836481394621267295993117142369, 2.02760841365534393785105269913, 2.92282612118883291678072368072, 4.18792924045637888091612635364, 4.87192026932947252345106588910, 6.24023653694325247163988929422, 7.56603176206504646875266770371, 8.422664636876819308152657515585, 8.928219778410772529161022570581, 9.711486156444590411319784609202

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