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Properties

Label 2-363-33.8-c1-0-7
Degree $2$
Conductor $363$
Sign $0.905 - 0.425i$
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  + (1.21 − 0.885i)2-s + (−1.43 − 0.970i)3-s + (0.0828 − 0.254i)4-s + (−1.71 + 2.35i)5-s + (−2.60 + 0.0882i)6-s + (2.68 + 0.874i)7-s + (0.806 + 2.48i)8-s + (1.11 + 2.78i)9-s + 4.38i·10-s + (−0.366 + 0.285i)12-s + (−0.526 − 0.725i)13-s + (4.05 − 1.31i)14-s + (4.73 − 1.71i)15-s + (3.61 + 2.62i)16-s + (2.11 + 1.53i)17-s + (3.82 + 2.40i)18-s + ⋯
L(s)  = 1  + (0.861 − 0.625i)2-s + (−0.828 − 0.560i)3-s + (0.0414 − 0.127i)4-s + (−0.764 + 1.05i)5-s + (−1.06 + 0.0360i)6-s + (1.01 + 0.330i)7-s + (0.284 + 0.877i)8-s + (0.372 + 0.927i)9-s + 1.38i·10-s + (−0.105 + 0.0823i)12-s + (−0.146 − 0.201i)13-s + (1.08 − 0.351i)14-s + (1.22 − 0.443i)15-s + (0.902 + 0.655i)16-s + (0.511 + 0.371i)17-s + (0.901 + 0.566i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.425i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 - 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.905 - 0.425i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.905 - 0.425i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42355 + 0.317630i\)
\(L(\frac12)\) \(\approx\) \(1.42355 + 0.317630i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.43 + 0.970i)T \)
11 \( 1 \)
good2 \( 1 + (-1.21 + 0.885i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (1.71 - 2.35i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-2.68 - 0.874i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (0.526 + 0.725i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-2.11 - 1.53i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.32 - 0.756i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (2.66 - 8.20i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-8.24 + 5.99i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (1.09 - 3.36i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.39 + 4.29i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 + (2.02 - 0.658i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.87 + 5.33i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.542 - 0.176i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.98 + 8.23i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (-5.58 + 7.69i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (3.05 + 0.991i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-4.31 - 5.94i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-5.11 - 3.71i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 + (4.63 - 3.36i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.46986596528263247968842098130, −11.11674681199216145391284313008, −10.27017459781095199179929793639, −8.272465455117097528043640558075, −7.74076185746311308046684583625, −6.61424420842524566504135902661, −5.41216357705788644784310013576, −4.51120099465608043663388292166, −3.26860522264468263679873183481, −1.94178264188570406175563071005, 0.921322746428814126718390548144, 3.92287265265311189523963721679, 4.63631293422266651114631100344, 5.15637147807408890951124574538, 6.27369761425825215615789173941, 7.41991956116645393061313592194, 8.419893880428544296655565073145, 9.613700588252942996698889070820, 10.56388592917557736471717435706, 11.63832280130295636111172748972

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