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Properties

Label 2-1815-165.104-c0-0-3
Degree $2$
Conductor $1815$
Sign $-0.923 - 0.382i$
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  + (0.535 + 1.64i)2-s + (0.809 + 0.587i)3-s + (−1.61 + 1.17i)4-s + (0.309 − 0.951i)5-s + (−0.535 + 1.64i)6-s + (−1.40 − 1.01i)8-s + (0.309 + 0.951i)9-s + 1.73·10-s − 2·12-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (−0.535 + 1.64i)17-s + (−1.40 + 1.01i)18-s + (0.618 + 1.90i)20-s + 23-s + (−0.535 − 1.64i)24-s + ⋯
L(s)  = 1  + (0.535 + 1.64i)2-s + (0.809 + 0.587i)3-s + (−1.61 + 1.17i)4-s + (0.309 − 0.951i)5-s + (−0.535 + 1.64i)6-s + (−1.40 − 1.01i)8-s + (0.309 + 0.951i)9-s + 1.73·10-s − 2·12-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (−0.535 + 1.64i)17-s + (−1.40 + 1.01i)18-s + (0.618 + 1.90i)20-s + 23-s + (−0.535 − 1.64i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-0.923 - 0.382i$
Analytic conductor: \(0.905802\)
Root analytic conductor: \(0.951736\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :0),\ -0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.857118678\)
\(L(\frac12)\) \(\approx\) \(1.857118678\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-0.535 - 1.64i)T + (-0.809 + 0.587i)T^{2} \)
7 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.535 - 1.64i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.535 + 1.64i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.535 + 1.64i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (-0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−9.314005779849021234843918916898, −8.763734279134567736726143618104, −8.217415218671791964751048329990, −7.55984875421439696565619117876, −6.54208094713616025033022716379, −5.72980743888037656104507117587, −4.97589579020083206326588579799, −4.28685672180684515957588240419, −3.57420621060066205574060693253, −1.97591358580546051465031042557, 1.20873786085515939217120398307, 2.43816103374506605957109761483, 2.82047950132424410147713282942, 3.65741771849464972727118255861, 4.66764074677438599133585514521, 5.71019950575326746720396149637, 6.92771655738706425327503590801, 7.35702753424612828922623798131, 8.747863501192013584990852543842, 9.306930685475737456956931361758

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