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Properties

Label 2-363-33.26-c2-0-10
Degree $2$
Conductor $363$
Sign $-0.782 - 0.622i$
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  + (−1.94 + 2.68i)2-s + (0.927 − 2.85i)3-s + (−2.16 − 6.65i)4-s + (3.89 + 5.36i)5-s + (5.84 + 8.04i)6-s + (2.47 + 7.60i)7-s + (9.46 + 3.07i)8-s + (−7.28 − 5.29i)9-s − 22·10-s − 21.0·12-s + (3.23 + 2.35i)13-s + (−25.2 − 8.19i)14-s + (18.9 − 6.14i)15-s + (−4.04 + 2.93i)16-s + (7.79 + 10.7i)17-s + (28.3 − 9.22i)18-s + ⋯
L(s)  = 1  + (−0.974 + 1.34i)2-s + (0.309 − 0.951i)3-s + (−0.540 − 1.66i)4-s + (0.779 + 1.07i)5-s + (0.974 + 1.34i)6-s + (0.353 + 1.08i)7-s + (1.18 + 0.384i)8-s + (−0.809 − 0.587i)9-s − 2.20·10-s − 1.75·12-s + (0.248 + 0.180i)13-s + (−1.80 − 0.585i)14-s + (1.26 − 0.409i)15-s + (−0.252 + 0.183i)16-s + (0.458 + 0.631i)17-s + (1.57 − 0.512i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 - 0.622i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.782 - 0.622i$
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: \((2,\ 363,\ (\ :1),\ -0.782 - 0.622i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.363195 + 1.03946i\)
\(L(\frac12)\) \(\approx\) \(0.363195 + 1.03946i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.927 + 2.85i)T \)
11 \( 1 \)
good2 \( 1 + (1.94 - 2.68i)T + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (-3.89 - 5.36i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (-2.47 - 7.60i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (-3.23 - 2.35i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (-7.79 - 10.7i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-1.85 + 5.70i)T + (-292. - 212. i)T^{2} \)
23 \( 1 - 6.63iT - 529T^{2} \)
29 \( 1 + (37.8 - 12.2i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-21.0 - 15.2i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-9.27 - 28.5i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (12.6 + 4.09i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 42T + 1.84e3T^{2} \)
47 \( 1 + (-82.0 - 26.6i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (35.0 - 48.2i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (63.0 - 20.4i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-9.70 + 7.05i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 2T + 4.48e3T^{2} \)
71 \( 1 + (-35.0 - 48.2i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-22.8 - 70.3i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (32.3 + 23.5i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (23.3 + 32.1i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + 119. iT - 7.92e3T^{2} \)
97 \( 1 + (50.1 + 36.4i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−11.46359303240686943167586981078, −10.33274257774617162120668209275, −9.312497065490889550222210981675, −8.638131552744511100147967625917, −7.74929802782485276633800743321, −6.88923440878658756990962244732, −6.12004921567806899377912604466, −5.52086809347233388583669667195, −2.92942368087034259985950863045, −1.61164901979040926132957937197, 0.68714535341500400824359264860, 2.00815729735403313241855426868, 3.47584398137650241764381898772, 4.48105422986129119929578883854, 5.62462033517108313960776636753, 7.68204334501734128972896432838, 8.503792814316979747493259178764, 9.391245908653827954914892031294, 9.849676745001051533943989668844, 10.66195919824824572330397604902

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