Properties

Label 2-297-11.3-c1-0-7
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: $\chi_{297} (190, \cdot )$
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

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

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