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Properties

Label 2-297-11.3-c1-0-12
Degree $2$
Conductor $297$
Sign $-0.370 + 0.928i$
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  + (0.255 − 0.185i)2-s + (−0.587 + 1.80i)4-s + (−3.28 − 2.38i)5-s + (1.05 − 3.24i)7-s + (0.380 + 1.17i)8-s − 1.28·10-s + (1.15 − 3.10i)11-s + (−1.11 + 0.812i)13-s + (−0.333 − 1.02i)14-s + (−2.75 − 2.00i)16-s + (−5.20 − 3.78i)17-s + (−1.07 − 3.31i)19-s + (6.23 − 4.52i)20-s + (−0.282 − 1.00i)22-s + 3.73·23-s + ⋯
L(s)  = 1  + (0.180 − 0.131i)2-s + (−0.293 + 0.903i)4-s + (−1.46 − 1.06i)5-s + (0.398 − 1.22i)7-s + (0.134 + 0.414i)8-s − 0.405·10-s + (0.348 − 0.937i)11-s + (−0.310 + 0.225i)13-s + (−0.0890 − 0.274i)14-s + (−0.689 − 0.501i)16-s + (−1.26 − 0.916i)17-s + (−0.247 − 0.760i)19-s + (1.39 − 1.01i)20-s + (−0.0601 − 0.215i)22-s + 0.779·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $-0.370 + 0.928i$
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: \((2,\ 297,\ (\ :1/2),\ -0.370 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.442083 - 0.652360i\)
\(L(\frac12)\) \(\approx\) \(0.442083 - 0.652360i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.15 + 3.10i)T \)
good2 \( 1 + (-0.255 + 0.185i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (3.28 + 2.38i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-1.05 + 3.24i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (1.11 - 0.812i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.20 + 3.78i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.07 + 3.31i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.73T + 23T^{2} \)
29 \( 1 + (0.0230 - 0.0709i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.02 + 2.19i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.381 - 1.17i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.87 - 8.86i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 0.456T + 43T^{2} \)
47 \( 1 + (0.678 + 2.08i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.729 + 0.530i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.47 - 7.62i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (4.66 + 3.39i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.85T + 67T^{2} \)
71 \( 1 + (-7.66 - 5.56i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.34 + 13.3i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-8.89 + 6.45i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.10 - 0.804i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + (-6.67 + 4.84i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.37567257041052651891782557113, −11.13968836281842264801272167397, −9.215800023835650873964172023928, −8.531656382294745394799553717275, −7.70673172466695109659237241964, −6.94301389352079722509747302554, −4.66622515238631210046230594471, −4.42669337511920998617046727051, −3.22993498996446771322827741566, −0.55933949770709824879660703695, 2.24024327835001354150175975453, 3.90154601735843900043666293437, 4.89022601363125421685095989263, 6.22375153662018340057868595157, 7.06937861424023731384384901992, 8.227508942635496335279574234034, 9.155692812811585487220941314234, 10.42004569524104873259785257987, 11.06017413745251954586770480049, 12.01062412501446899787931303852

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