Properties

Label 2-297-297.76-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  + (0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (−0.326 + 0.118i)5-s + (0.173 + 0.984i)9-s + (−0.939 − 0.342i)11-s + (0.766 − 0.642i)12-s + (−0.326 − 0.118i)15-s + (−0.939 − 0.342i)16-s + (0.0603 + 0.342i)20-s + (−0.173 + 0.984i)23-s + (−0.673 + 0.565i)25-s + (−0.500 + 0.866i)27-s + (0.266 − 1.50i)31-s + (−0.5 − 0.866i)33-s + 36-s + (−0.173 + 0.300i)37-s + ⋯
L(s)  = 1  + (0.766 + 0.642i)3-s + (0.173 − 0.984i)4-s + (−0.326 + 0.118i)5-s + (0.173 + 0.984i)9-s + (−0.939 − 0.342i)11-s + (0.766 − 0.642i)12-s + (−0.326 − 0.118i)15-s + (−0.939 − 0.342i)16-s + (0.0603 + 0.342i)20-s + (−0.173 + 0.984i)23-s + (−0.673 + 0.565i)25-s + (−0.500 + 0.866i)27-s + (0.266 − 1.50i)31-s + (−0.5 − 0.866i)33-s + 36-s + (−0.173 + 0.300i)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: $\chi_{297} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :0),\ 0.998 - 0.0581i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9080947948\)
\(L(\frac12)\) \(\approx\) \(0.9080947948\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.766 - 0.642i)T \)
11 \( 1 + (0.939 + 0.342i)T \)
good2 \( 1 + (-0.173 + 0.984i)T^{2} \)
5 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
7 \( 1 + (0.939 - 0.342i)T^{2} \)
13 \( 1 + (-0.173 - 0.984i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
29 \( 1 + (-0.173 + 0.984i)T^{2} \)
31 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
37 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \)
53 \( 1 - 1.53T + T^{2} \)
59 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (0.939 - 0.342i)T^{2} \)
67 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
71 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (-0.173 + 0.984i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
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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

−11.68454496604138985197082128615, −10.89710830963381586137116127701, −10.06024098828169117661847420476, −9.379752967256463840250575735044, −8.218698807691377846633469704155, −7.33529029505524823313294128049, −5.84471138623774864204276039501, −4.92723638224117684674520181114, −3.59681502528997047131938143207, −2.19370799218050459251125349540, 2.29619675943427701773647904468, 3.36417730849738865534218497816, 4.61173491729997113317236599889, 6.40881090642502171409726828797, 7.39431282861877647559838915823, 8.101897095820321120255472863980, 8.773741081934350896351335190754, 10.01923088151719031952640022397, 11.24431173327927600006780412288, 12.40928949119541438896602245903

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