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Properties

Label 2-297-11.4-c1-0-2
Degree $2$
Conductor $297$
Sign $0.992 - 0.124i$
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  + (−1.32 − 0.959i)2-s + (0.206 + 0.634i)4-s + (−2.42 + 1.76i)5-s + (−0.469 − 1.44i)7-s + (−0.672 + 2.07i)8-s + 4.89·10-s + (3.29 − 0.334i)11-s + (3.32 + 2.41i)13-s + (−0.766 + 2.35i)14-s + (3.95 − 2.87i)16-s + (2.16 − 1.57i)17-s + (−2.64 + 8.14i)19-s + (−1.61 − 1.17i)20-s + (−4.68 − 2.72i)22-s + 4.20·23-s + ⋯
L(s)  = 1  + (−0.934 − 0.678i)2-s + (0.103 + 0.317i)4-s + (−1.08 + 0.787i)5-s + (−0.177 − 0.546i)7-s + (−0.237 + 0.731i)8-s + 1.54·10-s + (0.994 − 0.100i)11-s + (0.921 + 0.669i)13-s + (−0.204 + 0.630i)14-s + (0.988 − 0.718i)16-s + (0.526 − 0.382i)17-s + (−0.607 + 1.86i)19-s + (−0.361 − 0.262i)20-s + (−0.997 − 0.580i)22-s + 0.876·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(297\)    =    \(3^{3} \cdot 11\)
Sign: $0.992 - 0.124i$
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: \((2,\ 297,\ (\ :1/2),\ 0.992 - 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.621643 + 0.0388692i\)
\(L(\frac12)\) \(\approx\) \(0.621643 + 0.0388692i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.29 + 0.334i)T \)
good2 \( 1 + (1.32 + 0.959i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (2.42 - 1.76i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.469 + 1.44i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-3.32 - 2.41i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.16 + 1.57i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.64 - 8.14i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 4.20T + 23T^{2} \)
29 \( 1 + (0.598 + 1.84i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-5.57 - 4.04i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.208 + 0.643i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.04 - 6.29i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 0.0153T + 43T^{2} \)
47 \( 1 + (-1.19 + 3.67i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (0.0180 + 0.0130i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.17 - 12.8i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (9.51 - 6.91i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 7.15T + 67T^{2} \)
71 \( 1 + (2.70 - 1.96i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.56 + 7.89i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.38 - 3.18i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.82 - 6.41i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 9.84T + 89T^{2} \)
97 \( 1 + (-12.7 - 9.24i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.66294398517976617898911402014, −10.72949419177089526839737184606, −10.13108783370154609149559430015, −8.982469914094000867179421505529, −8.159227934627136003779417339385, −7.14191071866206320151169993984, −6.06770878546604173222040510486, −4.15834887172726813022899825836, −3.22015843015374313068912806504, −1.31016897861833856353267554772, 0.76085037870410335095491986232, 3.39933412877461030897059640340, 4.57245895801512562243503581136, 6.10063362686248934683715990831, 7.09291805720013984694289927726, 8.127964401000680136718198029351, 8.783858130023780505388205171802, 9.325357242291895660181096154338, 10.74924087928995561737991198351, 11.76698358495719298778496561227

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