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Properties

Label 2-363-11.4-c1-0-12
Degree $2$
Conductor $363$
Sign $0.659 - 0.751i$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
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  + (2.04 + 1.48i)2-s + (0.309 − 0.951i)3-s + (1.35 + 4.15i)4-s + (1.91 − 1.39i)5-s + (2.04 − 1.48i)6-s + (−0.244 − 0.753i)7-s + (−1.85 + 5.69i)8-s + (−0.809 − 0.587i)9-s + 5.98·10-s + 4.37·12-s + (−3.32 − 2.41i)13-s + (0.618 − 1.90i)14-s + (−0.733 − 2.25i)15-s + (−5.15 + 3.74i)16-s + (−4.84 + 3.51i)17-s + (−0.780 − 2.40i)18-s + ⋯
L(s)  = 1  + (1.44 + 1.04i)2-s + (0.178 − 0.549i)3-s + (0.675 + 2.07i)4-s + (0.858 − 0.623i)5-s + (0.833 − 0.605i)6-s + (−0.0925 − 0.284i)7-s + (−0.654 + 2.01i)8-s + (−0.269 − 0.195i)9-s + 1.89·10-s + 1.26·12-s + (−0.921 − 0.669i)13-s + (0.165 − 0.508i)14-s + (−0.189 − 0.582i)15-s + (−1.28 + 0.936i)16-s + (−1.17 + 0.853i)17-s + (−0.183 − 0.565i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.659 - 0.751i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (202, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.84968 + 1.29114i\)
\(L(\frac12)\) \(\approx\) \(2.84968 + 1.29114i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-2.04 - 1.48i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-1.91 + 1.39i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.244 + 0.753i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.32 + 2.41i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (4.84 - 3.51i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.31 - 4.04i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + (-0.780 - 2.40i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.96 + 4.33i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-1.54 - 4.75i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.75 + 5.41i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 6.63T + 43T^{2} \)
47 \( 1 + (0.387 - 1.19i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (10.6 + 7.70i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.85 + 5.70i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.16 - 1.57i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 16.1T + 67T^{2} \)
71 \( 1 + (0.602 - 0.437i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.29 + 7.06i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.72 - 3.43i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-6.88 + 5.00i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 6.37T + 89T^{2} \)
97 \( 1 + (-10.1 - 7.34i)T + (29.9 + 92.2i)T^{2} \)
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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

−12.19288765028738570115195504986, −10.84771112839467234918442898736, −9.507086387558719661938609624344, −8.363472265141825207909613869262, −7.50090382129395780465298103434, −6.53483924294312865866600443418, −5.73585399959510680874266753330, −4.88395399425372210510049681943, −3.68043327897373809642504144276, −2.17223613167512368719926516358, 2.25003971314852961356507237130, 2.80196072639219902061330052493, 4.27922432912401501028706396725, 5.03959251819577741238894872750, 6.11467717295028886089627366000, 7.05433935748048453147273445344, 9.142831784581604241495592782077, 9.676402940056464003672878268581, 10.80339735622160562169578449168, 11.18767086752607706112717163110

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