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Properties

Label 2-297-297.175-c0-0-0
Degree $2$
Conductor $297$
Sign $-0.448 + 0.893i$
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  + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (−1.43 − 1.20i)5-s + (−0.939 − 0.342i)9-s + (0.766 − 0.642i)11-s + (0.173 + 0.984i)12-s + (−1.43 + 1.20i)15-s + (0.766 − 0.642i)16-s + (1.76 + 0.642i)20-s + (0.939 − 0.342i)23-s + (0.439 + 2.49i)25-s + (−0.5 + 0.866i)27-s + (−0.326 + 0.118i)31-s + (−0.5 − 0.866i)33-s + 0.999·36-s + (0.939 − 1.62i)37-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)3-s + (−0.939 + 0.342i)4-s + (−1.43 − 1.20i)5-s + (−0.939 − 0.342i)9-s + (0.766 − 0.642i)11-s + (0.173 + 0.984i)12-s + (−1.43 + 1.20i)15-s + (0.766 − 0.642i)16-s + (1.76 + 0.642i)20-s + (0.939 − 0.342i)23-s + (0.439 + 2.49i)25-s + (−0.5 + 0.866i)27-s + (−0.326 + 0.118i)31-s + (−0.5 − 0.866i)33-s + 0.999·36-s + (0.939 − 1.62i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.448 + 0.893i$
Analytic conductor: \(0.148222\)
Root analytic conductor: \(0.384996\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :0),\ -0.448 + 0.893i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5063821624\)
\(L(\frac12)\) \(\approx\) \(0.5063821624\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.173 + 0.984i)T \)
11 \( 1 + (-0.766 + 0.642i)T \)
good2 \( 1 + (0.939 - 0.342i)T^{2} \)
5 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (-0.766 - 0.642i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
37 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \)
53 \( 1 - 0.347T + T^{2} \)
59 \( 1 + (-0.266 - 0.223i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.766 - 1.32i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.939 - 0.342i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.17 + 0.984i)T + (0.173 - 0.984i)T^{2} \)
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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

−11.91396847228151103016869557294, −11.19695650726369249432850093526, −9.220039328163842485012296009008, −8.715545822993928387111914515755, −8.008646794304444163677104053839, −7.14468547815721637488562187364, −5.55918773180648514914797225710, −4.36598838527626367752588302057, −3.40459537878854720293399672875, −0.864307895157398535155276217590, 3.16794659852556745005572638295, 4.04047450003798566291325691540, 4.86139087454397805016920937725, 6.43120060138251007139008625511, 7.63821013592983897437007039599, 8.588258130273881292479443876364, 9.579274651870016506474292225442, 10.38469081680986122858317922659, 11.25171570023346297148621185333, 11.96584597253416399555273709629

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