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Properties

Label 2-1815-165.14-c0-0-2
Degree $2$
Conductor $1815$
Sign $-0.975 - 0.220i$
Analytic cond. $0.905802$
Root an. cond. $0.951736$
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  + (−1.40 + 1.01i)2-s + (0.309 + 0.951i)3-s + (0.618 − 1.90i)4-s + (0.809 + 0.587i)5-s + (−1.40 − 1.01i)6-s + (0.535 + 1.64i)8-s + (−0.809 + 0.587i)9-s − 1.73·10-s + 1.99·12-s + (−0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (1.40 + 1.01i)17-s + (0.535 − 1.64i)18-s + (1.61 − 1.17i)20-s − 23-s + (−1.40 + 1.01i)24-s + ⋯
L(s)  = 1  + (−1.40 + 1.01i)2-s + (0.309 + 0.951i)3-s + (0.618 − 1.90i)4-s + (0.809 + 0.587i)5-s + (−1.40 − 1.01i)6-s + (0.535 + 1.64i)8-s + (−0.809 + 0.587i)9-s − 1.73·10-s + 1.99·12-s + (−0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (1.40 + 1.01i)17-s + (0.535 − 1.64i)18-s + (1.61 − 1.17i)20-s − 23-s + (−1.40 + 1.01i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 - 0.220i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.975 - 0.220i$
Analytic conductor: \(0.905802\)
Root analytic conductor: \(0.951736\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1334, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :0),\ -0.975 - 0.220i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6948440525\)
\(L(\frac12)\) \(\approx\) \(0.6948440525\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (1.40 - 1.01i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-1.40 - 1.01i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.809 + 0.587i)T^{2} \)
23 \( 1 + T + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-1.40 - 1.01i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (1.40 - 1.01i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−9.861304230240334256324975952758, −9.090674415241393554575890283954, −8.210948247248275847254615868715, −7.79421769700450664696015339082, −6.71133917724396828483036909365, −5.89397186996623508565870842861, −5.47582534441819380678715205903, −4.06976699347652051559024222819, −2.87668157304938498715555671690, −1.61387248070051777455999246556, 0.842111057229030283016995287525, 1.74095648868573354325946318003, 2.60061573755272976887573038500, 3.47583072145244819800611256103, 5.09953563008953531685464562589, 6.08844883332088387479254727283, 7.06536505448535718743914150712, 7.910950233041219689386774952785, 8.388271877850161720679968004988, 9.182618328409140022838760918962

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