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Properties

Label 2-1815-165.14-c0-0-6
Degree $2$
Conductor $1815$
Sign $0.751 + 0.659i$
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  + (0.809 − 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.809 + 0.587i)5-s + (−0.809 − 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + 10-s + (0.309 − 0.951i)15-s + (0.809 + 0.587i)16-s + (0.809 + 0.587i)17-s + (−0.309 + 0.951i)18-s + (−0.618 − 1.90i)19-s + 23-s + (0.809 − 0.587i)24-s + (0.309 + 0.951i)25-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)2-s + (−0.309 − 0.951i)3-s + (0.809 + 0.587i)5-s + (−0.809 − 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + 10-s + (0.309 − 0.951i)15-s + (0.809 + 0.587i)16-s + (0.809 + 0.587i)17-s + (−0.309 + 0.951i)18-s + (−0.618 − 1.90i)19-s + 23-s + (0.809 − 0.587i)24-s + (0.309 + 0.951i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $0.751 + 0.659i$
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: \((2,\ 1815,\ (\ :0),\ 0.751 + 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.758485630\)
\(L(\frac12)\) \(\approx\) \(1.758485630\)
\(L(1)\) 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 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.618 + 1.90i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
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

−9.366242352849773932949638787867, −8.517809905539619219160737617213, −7.65554323058077395117503885367, −6.84387543022908256113296608908, −6.11359059655095483750559467364, −5.29594569416410271093364630038, −4.50293959135026576152128996879, −3.07130135297649411921477437715, −2.60037561316885179311880702467, −1.50922481275691110329713552696, 1.32249093631064890894056120376, 3.05638873480204012376587505711, 4.04850707253205857848408679833, 4.83510814530480508738061643693, 5.45617372283599861474287872129, 6.01877532595575012834644629535, 6.77168807336329628159628127579, 8.012244022417512299841467161312, 8.884735211855961110744284800559, 9.743414506306832351203404303184

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