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Properties

Label 2-363-33.29-c1-0-19
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: $\chi_{363} (161, \cdot )$
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

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

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