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Properties

Label 2-363-11.3-c1-0-3
Degree $2$
Conductor $363$
Sign $-0.751 - 0.659i$
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  + (−0.5 + 0.363i)2-s + (0.309 + 0.951i)3-s + (−0.5 + 1.53i)4-s + (2.11 + 1.53i)5-s + (−0.5 − 0.363i)6-s + (−0.927 + 2.85i)7-s + (−0.690 − 2.12i)8-s + (−0.809 + 0.587i)9-s − 1.61·10-s − 1.61·12-s + (1.42 − 1.03i)13-s + (−0.572 − 1.76i)14-s + (−0.809 + 2.48i)15-s + (−1.49 − 1.08i)16-s + (1.30 + 0.951i)17-s + (0.190 − 0.587i)18-s + ⋯
L(s)  = 1  + (−0.353 + 0.256i)2-s + (0.178 + 0.549i)3-s + (−0.250 + 0.769i)4-s + (0.947 + 0.688i)5-s + (−0.204 − 0.148i)6-s + (−0.350 + 1.07i)7-s + (−0.244 − 0.751i)8-s + (−0.269 + 0.195i)9-s − 0.511·10-s − 0.467·12-s + (0.395 − 0.287i)13-s + (−0.153 − 0.471i)14-s + (−0.208 + 0.642i)15-s + (−0.374 − 0.272i)16-s + (0.317 + 0.230i)17-s + (0.0450 − 0.138i)18-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.751 - 0.659i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.412995 + 1.09748i\)
\(L(\frac12)\) \(\approx\) \(0.412995 + 1.09748i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.363i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-2.11 - 1.53i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.927 - 2.85i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.42 + 1.03i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.30 - 0.951i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.80 + 5.56i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 + (1.38 - 4.25i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (2.30 - 1.67i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.0729 + 0.224i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.69 - 11.3i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.23T + 43T^{2} \)
47 \( 1 + (-0.5 - 1.53i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.78 + 5.65i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.19 + 9.82i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.35 - 4.61i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 9.56T + 67T^{2} \)
71 \( 1 + (-4.5 - 3.26i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1 + 3.07i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-7.66 + 5.56i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.572 - 0.416i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 0.527T + 89T^{2} \)
97 \( 1 + (-11.3 + 8.24i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.68738350475949827684276087777, −10.70122351442075975151750186283, −9.694917092260331964034640553916, −9.058789271319819548418828106978, −8.333120030424266702211878757052, −6.99193715476250176634363956813, −6.12542947214131653677647841175, −4.97070354680024222411578640611, −3.38454046814789050640293100230, −2.53850750597214293231704952973, 0.928664665859263702013630722984, 2.02375002024863075118287765577, 3.94664202589201957480362358997, 5.37469693186460567870880355434, 6.14463405279921514306950701578, 7.29120252749070145096650346640, 8.522039856768935507812202544591, 9.329921962513959354053154094587, 10.09072182437223029648846060275, 10.80064310481753265910553532732

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