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Properties

Label 2-363-33.26-c2-0-31
Degree $2$
Conductor $363$
Sign $0.975 - 0.218i$
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  + (0.399 − 0.550i)2-s + (2.97 − 0.384i)3-s + (1.09 + 3.36i)4-s + (−3.81 − 5.25i)5-s + (0.978 − 1.79i)6-s + (2.89 + 8.90i)7-s + (4.87 + 1.58i)8-s + (8.70 − 2.28i)9-s − 4.41·10-s + (4.54 + 9.58i)12-s + (3.37 + 2.45i)13-s + (6.06 + 1.96i)14-s + (−13.3 − 14.1i)15-s + (−8.62 + 6.26i)16-s + (17.5 + 24.2i)17-s + (2.22 − 5.70i)18-s + ⋯
L(s)  = 1  + (0.199 − 0.275i)2-s + (0.991 − 0.128i)3-s + (0.273 + 0.840i)4-s + (−0.762 − 1.05i)5-s + (0.163 − 0.298i)6-s + (0.413 + 1.27i)7-s + (0.609 + 0.198i)8-s + (0.967 − 0.253i)9-s − 0.441·10-s + (0.378 + 0.799i)12-s + (0.259 + 0.188i)13-s + (0.432 + 0.140i)14-s + (−0.891 − 0.943i)15-s + (−0.538 + 0.391i)16-s + (1.03 + 1.42i)17-s + (0.123 − 0.317i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.975 - 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.975 - 0.218i$
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: \((2,\ 363,\ (\ :1),\ 0.975 - 0.218i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.69814 + 0.297865i\)
\(L(\frac12)\) \(\approx\) \(2.69814 + 0.297865i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.97 + 0.384i)T \)
11 \( 1 \)
good2 \( 1 + (-0.399 + 0.550i)T + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (3.81 + 5.25i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (-2.89 - 8.90i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (-3.37 - 2.45i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (-17.5 - 24.2i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-3.18 + 9.78i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + 27.8iT - 529T^{2} \)
29 \( 1 + (-15.3 + 4.99i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-14.2 - 10.3i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-4.49 - 13.8i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (33.2 + 10.8i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 64.3T + 1.84e3T^{2} \)
47 \( 1 + (9.94 + 3.23i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-35.5 + 48.8i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (38.6 - 12.5i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (19.0 - 13.8i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 30.8T + 4.48e3T^{2} \)
71 \( 1 + (-0.585 - 0.805i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (28.8 + 88.8i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-91.4 - 66.4i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (61.1 + 84.1i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 - 20.3iT - 7.92e3T^{2} \)
97 \( 1 + (153. + 111. i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−11.70776695577994571078514998463, −10.32488458682521655507088714561, −8.930571418801947226311237424046, −8.319856893696727087860603313741, −8.059723721006562175656706674930, −6.62139975478499383959260631518, −4.97907387288293204304879472269, −4.01717439250624861508619361068, −2.93450628980261249443409764262, −1.66586996470975010047573926690, 1.24691299760903864214697990035, 3.02054461050753820854195972009, 3.96092752877634040620570403667, 5.15201151463478966536710207463, 6.75985670948956957576018071413, 7.40772910093026427022972968119, 7.967111893195468041533837971384, 9.614570301229940960120776531949, 10.23054079743524312440416224368, 10.99851113586778342177828801954

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