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Properties

Label 2-363-33.26-c2-0-8
Degree $2$
Conductor $363$
Sign $-0.696 - 0.717i$
Analytic cond. $9.89103$
Root an. cond. $3.14500$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.10 + 2.90i)2-s + (−2.00 + 2.23i)3-s + (−2.74 − 8.43i)4-s + (1.22 + 1.68i)5-s + (−2.26 − 10.5i)6-s + (−2.73 − 8.42i)7-s + (16.6 + 5.39i)8-s + (−0.979 − 8.94i)9-s − 7.48·10-s + (24.3 + 10.7i)12-s + (−1.33 − 0.966i)13-s + (30.2 + 9.82i)14-s + (−6.22 − 0.640i)15-s + (−21.9 + 15.9i)16-s + (7.30 + 10.0i)17-s + (28.0 + 16.0i)18-s + ⋯
L(s)  = 1  + (−1.05 + 1.45i)2-s + (−0.667 + 0.744i)3-s + (−0.685 − 2.10i)4-s + (0.245 + 0.337i)5-s + (−0.376 − 1.75i)6-s + (−0.391 − 1.20i)7-s + (2.07 + 0.674i)8-s + (−0.108 − 0.994i)9-s − 0.748·10-s + (2.02 + 0.897i)12-s + (−0.102 − 0.0743i)13-s + (2.15 + 0.701i)14-s + (−0.415 − 0.0427i)15-s + (−1.37 + 0.998i)16-s + (0.429 + 0.591i)17-s + (1.55 + 0.890i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.696 - 0.717i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ -0.696 - 0.717i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.239814 + 0.566902i\)
\(L(\frac12)\) \(\approx\) \(0.239814 + 0.566902i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.00 - 2.23i)T \)
11 \( 1 \)
good2 \( 1 + (2.10 - 2.90i)T + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (-1.22 - 1.68i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (2.73 + 8.42i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (1.33 + 0.966i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (-7.30 - 10.0i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (3.26 - 10.0i)T + (-292. - 212. i)T^{2} \)
23 \( 1 - 20.3iT - 529T^{2} \)
29 \( 1 + (-11.0 + 3.58i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-18.9 - 13.7i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-2.23 - 6.89i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (36.9 + 12.0i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 - 15.8T + 1.84e3T^{2} \)
47 \( 1 + (-43.0 - 14.0i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-23.6 + 32.6i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (107. - 34.9i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-62.7 + 45.5i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 62.9T + 4.48e3T^{2} \)
71 \( 1 + (-6.00 - 8.26i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-23.0 - 70.9i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-69.8 - 50.7i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-6.94 - 9.55i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 74.5iT - 7.92e3T^{2} \)
97 \( 1 + (-62.4 - 45.3i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−10.96784468868151672397548182973, −10.12906939895299649178290308001, −9.935906520535253280714058403453, −8.732221511659495874830934085962, −7.69239190569433459821794927835, −6.73543810524868211864360637512, −6.11414914464184254189863520455, −5.04296382546782469289033802708, −3.76401024955057987435010475095, −0.860976019529524374877507430916, 0.64331446711368242763070139141, 2.03612154319808083085572565231, 2.94333932615584488191347913265, 4.83142761764234142600006984118, 6.05675281187504984314883319378, 7.34373191800972759569313355272, 8.470173633556479276222349291110, 9.116488885059735144853724947934, 10.03049499616857147533195210088, 10.97440463083351345346637338361

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