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Properties

Label 2-297-11.3-c1-0-13
Degree $2$
Conductor $297$
Sign $0.350 + 0.936i$
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  + (1.85 − 1.35i)2-s + (1.01 − 3.12i)4-s + (1.37 + 0.997i)5-s + (0.0641 − 0.197i)7-s + (−0.910 − 2.80i)8-s + 3.90·10-s + (−1.23 − 3.07i)11-s + (−1.11 + 0.812i)13-s + (−0.147 − 0.453i)14-s + (−0.168 − 0.122i)16-s + (−2.97 − 2.15i)17-s + (2.50 + 7.70i)19-s + (4.50 − 3.27i)20-s + (−6.45 − 4.05i)22-s − 3.99·23-s + ⋯
L(s)  = 1  + (1.31 − 0.955i)2-s + (0.507 − 1.56i)4-s + (0.613 + 0.446i)5-s + (0.0242 − 0.0746i)7-s + (−0.321 − 0.990i)8-s + 1.23·10-s + (−0.372 − 0.928i)11-s + (−0.310 + 0.225i)13-s + (−0.0394 − 0.121i)14-s + (−0.0420 − 0.0305i)16-s + (−0.720 − 0.523i)17-s + (0.574 + 1.76i)19-s + (1.00 − 0.732i)20-s + (−1.37 − 0.864i)22-s − 0.833·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.350 + 0.936i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ 0.350 + 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.15397 - 1.49449i\)
\(L(\frac12)\) \(\approx\) \(2.15397 - 1.49449i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.23 + 3.07i)T \)
good2 \( 1 + (-1.85 + 1.35i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.37 - 0.997i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.0641 + 0.197i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.11 - 0.812i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.97 + 2.15i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.50 - 7.70i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 3.99T + 23T^{2} \)
29 \( 1 + (0.167 - 0.515i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (7.95 - 5.77i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.381 - 1.17i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.88 + 8.87i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.92T + 43T^{2} \)
47 \( 1 + (-2.42 - 7.47i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-5.30 + 3.85i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.43 - 7.48i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.12 - 0.816i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.85T + 67T^{2} \)
71 \( 1 + (10.3 + 7.49i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.36 + 7.27i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.69 + 3.41i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (11.2 + 8.15i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.19T + 89T^{2} \)
97 \( 1 + (-5.68 + 4.12i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.79938846542205341743795416403, −10.68361083840570561824043847149, −10.29299603088405711762256628556, −8.980389194561914078370434353162, −7.54259832580056675642976539308, −6.08702211363083666682297537118, −5.49390809394920957370035289466, −4.14435678837940629000736164509, −3.07451244414256057759485608115, −1.90810117519678105774943179390, 2.39214183837543640507628896943, 4.05788206132539972428825187757, 5.02797489978215035364102751444, 5.75477804861109614053090449913, 6.91020327640847583930529814967, 7.65558976878979309286466422039, 9.009563229004637016417339189484, 9.958647325348700474653971076933, 11.31487771357301366388541934351, 12.38560308814472204513708015680

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