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Properties

Label 2-297-11.4-c1-0-12
Degree $2$
Conductor $297$
Sign $-0.323 + 0.946i$
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  + (0.293 + 0.212i)2-s + (−0.577 − 1.77i)4-s + (−2.17 + 1.57i)5-s + (−0.770 − 2.37i)7-s + (0.433 − 1.33i)8-s − 0.972·10-s + (1.14 − 3.11i)11-s + (−3.08 − 2.24i)13-s + (0.279 − 0.859i)14-s + (−2.61 + 1.89i)16-s + (0.909 − 0.660i)17-s + (2.09 − 6.45i)19-s + (4.06 + 2.95i)20-s + (0.998 − 0.667i)22-s − 6.74·23-s + ⋯
L(s)  = 1  + (0.207 + 0.150i)2-s + (−0.288 − 0.888i)4-s + (−0.971 + 0.705i)5-s + (−0.291 − 0.896i)7-s + (0.153 − 0.471i)8-s − 0.307·10-s + (0.346 − 0.938i)11-s + (−0.856 − 0.622i)13-s + (0.0746 − 0.229i)14-s + (−0.653 + 0.474i)16-s + (0.220 − 0.160i)17-s + (0.480 − 1.48i)19-s + (0.907 + 0.659i)20-s + (0.212 − 0.142i)22-s − 1.40·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.323 + 0.946i$
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: \((2,\ 297,\ (\ :1/2),\ -0.323 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.482700 - 0.675260i\)
\(L(\frac12)\) \(\approx\) \(0.482700 - 0.675260i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.14 + 3.11i)T \)
good2 \( 1 + (-0.293 - 0.212i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (2.17 - 1.57i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.770 + 2.37i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.08 + 2.24i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-0.909 + 0.660i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.09 + 6.45i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6.74T + 23T^{2} \)
29 \( 1 + (-2.29 - 7.06i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.42 - 3.94i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.0589 + 0.181i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (0.196 - 0.605i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.72T + 43T^{2} \)
47 \( 1 + (-3.48 + 10.7i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.45 - 3.23i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.60 + 8.01i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.39 - 1.01i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 4.50T + 67T^{2} \)
71 \( 1 + (9.95 - 7.22i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.658 + 2.02i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-9.37 - 6.80i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (0.662 - 0.481i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 1.42T + 89T^{2} \)
97 \( 1 + (8.14 + 5.91i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.30538155234206424204354506636, −10.56580935657607393855969665104, −9.832698798209462449556583475732, −8.583315560151230917011593923191, −7.33242386859503912909050247946, −6.71190033928293235542138412544, −5.39925748265348286615620491731, −4.21576631362894266449757197338, −3.10838998428248280975786276730, −0.57847245376936664191388544709, 2.36571328847657425199034984790, 3.95770454880061368757050036529, 4.56349468902745131666233614052, 6.02302363790838461238971379831, 7.60304565778190549968023947812, 8.065462678137117233501042138526, 9.165091516819487900778144332151, 9.959696129405552460970408947587, 11.79078943136695322940030747178, 12.11823567875113221027616515949

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