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Properties

Label 2-297-11.3-c1-0-6
Degree $2$
Conductor $297$
Sign $0.915 - 0.402i$
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  + (0.863 − 0.627i)2-s + (−0.266 + 0.819i)4-s + (0.993 + 0.721i)5-s + (−0.473 + 1.45i)7-s + (0.943 + 2.90i)8-s + 1.30·10-s + (3.00 − 1.40i)11-s + (−1.89 + 1.37i)13-s + (0.505 + 1.55i)14-s + (1.24 + 0.900i)16-s + (2.87 + 2.08i)17-s + (−0.884 − 2.72i)19-s + (−0.856 + 0.622i)20-s + (1.70 − 3.09i)22-s + 2.29·23-s + ⋯
L(s)  = 1  + (0.610 − 0.443i)2-s + (−0.133 + 0.409i)4-s + (0.444 + 0.322i)5-s + (−0.179 + 0.550i)7-s + (0.333 + 1.02i)8-s + 0.414·10-s + (0.905 − 0.424i)11-s + (−0.525 + 0.381i)13-s + (0.135 + 0.415i)14-s + (0.310 + 0.225i)16-s + (0.696 + 0.506i)17-s + (−0.202 − 0.624i)19-s + (−0.191 + 0.139i)20-s + (0.364 − 0.660i)22-s + 0.478·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.915 - 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.915 - 0.402i$
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: \((2,\ 297,\ (\ :1/2),\ 0.915 - 0.402i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75972 + 0.369874i\)
\(L(\frac12)\) \(\approx\) \(1.75972 + 0.369874i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.00 + 1.40i)T \)
good2 \( 1 + (-0.863 + 0.627i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-0.993 - 0.721i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.473 - 1.45i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.89 - 1.37i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.87 - 2.08i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.884 + 2.72i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.29T + 23T^{2} \)
29 \( 1 + (-2.56 + 7.88i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.91 - 2.84i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.24 + 3.82i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.49 - 4.60i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.52T + 43T^{2} \)
47 \( 1 + (2.82 + 8.70i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.00 - 3.63i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.36 + 10.3i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.496 - 0.360i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 - 8.65T + 67T^{2} \)
71 \( 1 + (12.6 + 9.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.68 - 14.4i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-1.57 + 1.14i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-4.31 - 3.13i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 5.84T + 89T^{2} \)
97 \( 1 + (-6.88 + 4.99i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.89895182030527187605697236246, −11.20698423834083205836612098292, −9.993444473073126256624387343996, −9.020222524927540277652201757039, −8.098703648956401584431282222245, −6.77345179640079087554757205447, −5.73636066562447015933901150252, −4.51018231565347367018349119103, −3.33976744184137729175085305618, −2.18322677911908228345977730754, 1.34617567256561080732162303261, 3.53385098241969456014372573394, 4.74381432849266624989444865003, 5.59721720494418298817244515383, 6.69166855328565252610587538030, 7.50624957205778191788307153096, 9.070913951843019383273982770457, 9.812047786699365830291655813477, 10.57221201488879496363363042147, 11.92752839442149772889114935482

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