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Properties

Label 2-297-297.274-c0-0-0
Degree $2$
Conductor $297$
Sign $0.998 - 0.0581i$
Analytic cond. $0.148222$
Root an. cond. $0.384996$
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  + (−0.939 + 0.342i)3-s + (0.766 + 0.642i)4-s + (0.266 − 1.50i)5-s + (0.766 − 0.642i)9-s + (0.173 + 0.984i)11-s + (−0.939 − 0.342i)12-s + (0.266 + 1.50i)15-s + (0.173 + 0.984i)16-s + (1.17 − 0.984i)20-s + (−0.766 − 0.642i)23-s + (−1.26 − 0.460i)25-s + (−0.500 + 0.866i)27-s + (−1.43 − 1.20i)31-s + (−0.5 − 0.866i)33-s + 36-s + (−0.766 + 1.32i)37-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)3-s + (0.766 + 0.642i)4-s + (0.266 − 1.50i)5-s + (0.766 − 0.642i)9-s + (0.173 + 0.984i)11-s + (−0.939 − 0.342i)12-s + (0.266 + 1.50i)15-s + (0.173 + 0.984i)16-s + (1.17 − 0.984i)20-s + (−0.766 − 0.642i)23-s + (−1.26 − 0.460i)25-s + (−0.500 + 0.866i)27-s + (−1.43 − 1.20i)31-s + (−0.5 − 0.866i)33-s + 36-s + (−0.766 + 1.32i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0581i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.998 - 0.0581i$
Analytic conductor: \(0.148222\)
Root analytic conductor: \(0.384996\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :0),\ 0.998 - 0.0581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7209046461\)
\(L(\frac12)\) \(\approx\) \(0.7209046461\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.939 - 0.342i)T \)
11 \( 1 + (-0.173 - 0.984i)T \)
good2 \( 1 + (-0.766 - 0.642i)T^{2} \)
5 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
7 \( 1 + (-0.173 + 0.984i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
37 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.766 + 0.642i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T^{2} \)
47 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
53 \( 1 + 1.87T + T^{2} \)
59 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (-1.76 + 0.642i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (-0.766 - 0.642i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \)
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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

−12.17282350942807011627121850095, −11.26052427562958624707274489718, −10.13944877077859299935484154972, −9.273285103350682363404845938377, −8.171775223482172798561658352019, −7.04979388009668601368832203048, −5.97799461313668822035095125698, −4.90346390341517136731147186619, −3.96739365909454404448995398838, −1.79216800080163308172253952172, 1.91061384467226185823412321544, 3.37820722857511416735701443745, 5.40461400541483976317311699937, 6.16493453451054474175619928489, 6.85423897090158580015044351358, 7.69037423958697255632119818374, 9.515996638389505812659223386882, 10.61168762527597610311657674663, 10.91368962105210014674517956821, 11.62506850157419188034548628419

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