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Properties

Label 2-297-11.3-c1-0-4
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

\(L(1)\) \(\approx\) \(1.14573 + 0.776426i\)
\(L(\frac12)\) \(\approx\) \(1.14573 + 0.776426i\)
\(L(\frac{3}{2})\) 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

−12.00539084192515528234126519251, −10.62544861359015423803535225342, −10.16323276990835218492452860361, −9.296095583068942608641514361546, −7.83552910068447910256997464935, −7.20475358145214724076150712038, −6.24758461505714361059999852249, −4.76398400857165841353270329726, −3.47209594565384658590024921408, −2.06757529255331104849732540979, 1.27579714151845046352576918571, 2.52395928967948534700226514426, 4.93331392648410045304354172803, 5.58197356312639605791064714049, 6.06507389225723041901060988047, 8.207277319438708709118186394996, 8.868336543523480546831345452813, 9.742398626219033209144462512077, 10.27047827005428249154030250575, 11.66494448803319944607545569725

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