Properties

Label 2-297-11.4-c1-0-5
Degree $2$
Conductor $297$
Sign $0.512 + 0.858i$
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  + (−1.89 − 1.37i)2-s + (1.07 + 3.30i)4-s + (2.90 − 2.10i)5-s + (0.355 + 1.09i)7-s + (1.07 − 3.29i)8-s − 8.39·10-s + (1.63 + 2.88i)11-s + (5.13 + 3.72i)13-s + (0.832 − 2.56i)14-s + (−0.931 + 0.676i)16-s + (−3.16 + 2.29i)17-s + (0.457 − 1.40i)19-s + (10.1 + 7.33i)20-s + (0.880 − 7.71i)22-s + 1.24·23-s + ⋯
L(s)  = 1  + (−1.33 − 0.972i)2-s + (0.537 + 1.65i)4-s + (1.29 − 0.943i)5-s + (0.134 + 0.413i)7-s + (0.378 − 1.16i)8-s − 2.65·10-s + (0.492 + 0.870i)11-s + (1.42 + 1.03i)13-s + (0.222 − 0.684i)14-s + (−0.232 + 0.169i)16-s + (−0.767 + 0.557i)17-s + (0.104 − 0.322i)19-s + (2.25 + 1.64i)20-s + (0.187 − 1.64i)22-s + 0.260·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.512 + 0.858i$
Analytic conductor: \(2.37155\)
Root analytic conductor: \(1.53998\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{297} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 297,\ (\ :1/2),\ 0.512 + 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.770759 - 0.437400i\)
\(L(\frac12)\) \(\approx\) \(0.770759 - 0.437400i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.63 - 2.88i)T \)
good2 \( 1 + (1.89 + 1.37i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-2.90 + 2.10i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.355 - 1.09i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-5.13 - 3.72i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (3.16 - 2.29i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.457 + 1.40i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.24T + 23T^{2} \)
29 \( 1 + (1.74 + 5.35i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.60 - 1.89i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (3.09 + 9.53i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-0.600 + 1.84i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.18T + 43T^{2} \)
47 \( 1 + (0.0767 - 0.236i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.71 - 3.42i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.04 - 6.28i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.66 + 1.93i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 14.0T + 67T^{2} \)
71 \( 1 + (-4.25 + 3.09i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.683 - 2.10i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.40 - 1.74i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (9.46 - 6.87i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 8.93T + 89T^{2} \)
97 \( 1 + (7.99 + 5.81i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.44805468062286164503195322776, −10.49147161082153487838788869093, −9.555294003252560337938384544759, −8.970259427292095867682820697707, −8.483627603237327033250423822533, −6.86116156194096188426972419083, −5.67772114609583971465981992240, −4.16804316408273162070495165439, −2.18791688534748463145088464474, −1.44937702349851252708058674982, 1.31934188392601830405943491718, 3.23517850085360378422868637545, 5.53799878991787514724200092332, 6.33685650389181304433326020199, 6.94733527864773119023274300766, 8.226782718276688763208897311648, 8.945228780258301653062444240598, 9.974682194774992174431052848515, 10.58855447134579159630248251510, 11.30921336892449853132115477799

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