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Properties

Label 2-1215-45.14-c0-0-4
Degree $2$
Conductor $1215$
Sign $0.173 - 0.984i$
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  + (0.766 + 1.32i)2-s + (−0.673 + 1.16i)4-s + (0.5 − 0.866i)5-s − 0.532·8-s + 1.53·10-s + (0.266 + 0.460i)16-s + 1.87·17-s − 1.87·19-s + (0.673 + 1.16i)20-s + (0.173 − 0.300i)23-s + (−0.499 − 0.866i)25-s + (−0.766 + 1.32i)31-s + (−0.673 + 1.16i)32-s + (1.43 + 2.49i)34-s + (−1.43 − 2.49i)38-s + ⋯
L(s)  = 1  + (0.766 + 1.32i)2-s + (−0.673 + 1.16i)4-s + (0.5 − 0.866i)5-s − 0.532·8-s + 1.53·10-s + (0.266 + 0.460i)16-s + 1.87·17-s − 1.87·19-s + (0.673 + 1.16i)20-s + (0.173 − 0.300i)23-s + (−0.499 − 0.866i)25-s + (−0.766 + 1.32i)31-s + (−0.673 + 1.16i)32-s + (1.43 + 2.49i)34-s + (−1.43 − 2.49i)38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1215\)    =    \(3^{5} \cdot 5\)
Sign: $0.173 - 0.984i$
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: \((2,\ 1215,\ (\ :0),\ 0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.699554443\)
\(L(\frac12)\) \(\approx\) \(1.699554443\)
\(L(1)\) 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 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - 1.87T + T^{2} \)
19 \( 1 + 1.87T + T^{2} \)
23 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + 0.347T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
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

−10.02916295591671874411463968394, −8.951086926373562067820260932797, −8.312483262613133812008220675804, −7.58474584808015885704007643643, −6.56996067873206129784059998510, −5.91401246889822356951081292231, −5.14503441570692434650679970820, −4.47767367291054647539826962080, −3.39654279676769974666416316783, −1.66709214923435557140922791929, 1.60691308174696087125175910576, 2.57063535871271363650588059072, 3.42730116950810653028055622756, 4.24986825529568919630353379582, 5.42403670455020529932128691637, 6.10412439818244328671598873798, 7.22932565649043497037887582754, 8.119963643353730992570877378681, 9.444627545998195666975216530512, 10.01594840273205174399204451795

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