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Properties

Label 2-1215-45.29-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: $\chi_{1215} (809, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1215,\ (\ :0),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1688037982\)
\(L(\frac12)\) \(\approx\) \(0.1688037982\)
\(L(1)\) 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

−9.231921328072860899869987795659, −8.759670695177023132922885939621, −8.240574202562754246406550029378, −7.34648656343247498910600371864, −6.56857693943562594530941649737, −5.82814440909629820915766987216, −4.73987223099865087273200788547, −4.01722195090846001145097387853, −2.14647678931587313488208304827, −0.17649504841220544373247996542, 1.89688052582169433740470844022, 2.68788986385754697996449066133, 3.73769458630966904567558343286, 4.53201975391963347938290264483, 6.20337346197717487868746486751, 6.85878814473584310651258680542, 7.988788250925296609113364052875, 8.723227559253335338813724677118, 9.362737673216622121173924614782, 10.41486090610659890306662384181

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