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Properties

Label 2-297-99.65-c1-0-7
Degree $2$
Conductor $297$
Sign $0.998 - 0.0561i$
Analytic cond. $2.37155$
Root an. cond. $1.53998$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)4-s + (3.68 − 2.12i)5-s + (−2.87 − 1.65i)11-s + (−1.99 + 3.46i)16-s + (7.37 + 4.25i)20-s + (−2.87 + 1.65i)23-s + (6.55 − 11.3i)25-s + (5.55 + 9.62i)31-s − 5.11·37-s − 6.63i·44-s + (−6.12 − 3.53i)47-s + (−3.5 − 6.06i)49-s + 1.43i·53-s − 14.1·55-s + (−9.81 + 5.66i)59-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (1.64 − 0.951i)5-s + (−0.866 − 0.500i)11-s + (−0.499 + 0.866i)16-s + (1.64 + 0.951i)20-s + (−0.598 + 0.345i)23-s + (1.31 − 2.27i)25-s + (0.998 + 1.72i)31-s − 0.841·37-s − 1.00i·44-s + (−0.893 − 0.516i)47-s + (−0.5 − 0.866i)49-s + 0.197i·53-s − 1.90·55-s + (−1.27 + 0.737i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.998 - 0.0561i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ 0.998 - 0.0561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.73409 + 0.0487610i\)
\(L(\frac12)\) \(\approx\) \(1.73409 + 0.0487610i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (2.87 + 1.65i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
5 \( 1 + (-3.68 + 2.12i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (2.87 - 1.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.55 - 9.62i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 5.11T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.12 + 3.53i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.43iT - 53T^{2} \)
59 \( 1 + (9.81 - 5.66i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.05 - 1.83i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.69iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 + (-8.55 + 14.8i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.00871464429938903010167334567, −10.70714949289943997611911090658, −9.918898337511144764699656429436, −8.796452325566257997652107630784, −8.166278873556751261167502675153, −6.77295871758263049816655760014, −5.78052714732256427149376560786, −4.80512946652009250519402585032, −3.04945806214358337515940235456, −1.79500035656915775258886071133, 1.88563412391320667062298837421, 2.75518159811523909875378913338, 4.96337395849272389746786223983, 5.97397549324412686872515571821, 6.54408859185573138809475425726, 7.68603430044049099713195368604, 9.370521731932759459136019742423, 10.03721883190934085280815018849, 10.55320828353993214271165253073, 11.46157835226140633042369967244

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