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Properties

Label 2-363-11.4-c1-0-0
Degree $2$
Conductor $363$
Sign $-0.927 - 0.374i$
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  + (−0.309 − 0.224i)2-s + (−0.309 + 0.951i)3-s + (−0.572 − 1.76i)4-s + (−1.30 + 0.951i)5-s + (0.309 − 0.224i)6-s + (0.309 + 0.951i)7-s + (−0.454 + 1.40i)8-s + (−0.809 − 0.587i)9-s + 0.618·10-s + 1.85·12-s + (−3.42 − 2.48i)13-s + (0.118 − 0.363i)14-s + (−0.499 − 1.53i)15-s + (−2.54 + 1.84i)16-s + (−6.35 + 4.61i)17-s + (0.118 + 0.363i)18-s + ⋯
L(s)  = 1  + (−0.218 − 0.158i)2-s + (−0.178 + 0.549i)3-s + (−0.286 − 0.881i)4-s + (−0.585 + 0.425i)5-s + (0.126 − 0.0916i)6-s + (0.116 + 0.359i)7-s + (−0.160 + 0.495i)8-s + (−0.269 − 0.195i)9-s + 0.195·10-s + 0.535·12-s + (−0.950 − 0.690i)13-s + (0.0315 − 0.0970i)14-s + (−0.129 − 0.397i)15-s + (−0.636 + 0.462i)16-s + (−1.54 + 1.11i)17-s + (0.0278 + 0.0856i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.927 - 0.374i$
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: \((2,\ 363,\ (\ :1/2),\ -0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0351981 + 0.181214i\)
\(L(\frac12)\) \(\approx\) \(0.0351981 + 0.181214i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.224i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (1.30 - 0.951i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.309 - 0.951i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (3.42 + 2.48i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (6.35 - 4.61i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.263 + 0.812i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.23T + 23T^{2} \)
29 \( 1 + (-1.85 - 5.70i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (4.11 + 2.99i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.545 + 1.67i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.30 + 4.02i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 6.70T + 43T^{2} \)
47 \( 1 + (-0.336 + 1.03i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.11 - 1.53i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.97 - 9.14i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.92 + 5.03i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 4.85T + 67T^{2} \)
71 \( 1 + (-4.30 + 3.13i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (2.38 + 7.33i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.89 - 6.46i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.04 - 4.39i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 3.76T + 89T^{2} \)
97 \( 1 + (0.927 + 0.673i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.52034388104432219905211337499, −10.78326622369111345983928462180, −10.17301973615062559601204260625, −9.158344559246405078350384819040, −8.360363740201522801495811347616, −7.05466099334094111026054064516, −5.88576372364379481721535902749, −4.97342856024598073191749865083, −3.84990407601444485678462766639, −2.21446844940188821761293792449, 0.13007060242265709801906984920, 2.42973866780133081041971408440, 4.06415307846894545888638803819, 4.83245424417012716936761698911, 6.58504836279860561466500977430, 7.32634544624290186172783555577, 8.123787834656179143775809837971, 8.958547749242082660914063438103, 9.938097001462012558312186704130, 11.47005960349031557036239492399

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