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Properties

Label 2-363-33.8-c1-0-22
Degree $2$
Conductor $363$
Sign $0.838 + 0.544i$
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  + (1.53 − 1.11i)2-s + (1.08 + 1.34i)3-s + (0.499 − 1.53i)4-s + (1.53 − 2.11i)5-s + (3.18 + 0.857i)6-s + (−0.690 − 0.224i)7-s + (0.224 + 0.690i)8-s + (−0.633 + 2.93i)9-s − 4.97i·10-s + (2.61 − 1.00i)12-s + (−1.80 − 2.48i)13-s + (−1.31 + 0.427i)14-s + (4.52 − 0.229i)15-s + (3.73 + 2.71i)16-s + (−2.12 − 1.54i)17-s + (2.30 + 5.22i)18-s + ⋯
L(s)  = 1  + (1.08 − 0.790i)2-s + (0.628 + 0.778i)3-s + (0.249 − 0.769i)4-s + (0.688 − 0.947i)5-s + (1.29 + 0.350i)6-s + (−0.261 − 0.0848i)7-s + (0.0793 + 0.244i)8-s + (−0.211 + 0.977i)9-s − 1.57i·10-s + (0.755 − 0.288i)12-s + (−0.501 − 0.690i)13-s + (−0.351 + 0.114i)14-s + (1.16 − 0.0593i)15-s + (0.934 + 0.678i)16-s + (−0.515 − 0.374i)17-s + (0.542 + 1.23i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.838 + 0.544i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.838 + 0.544i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.838 + 0.544i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.74967 - 0.814292i\)
\(L(\frac12)\) \(\approx\) \(2.74967 - 0.814292i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.08 - 1.34i)T \)
11 \( 1 \)
good2 \( 1 + (-1.53 + 1.11i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.53 + 2.11i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.690 + 0.224i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.80 + 2.48i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.12 + 1.54i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.04 - 1.31i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 1.76iT - 23T^{2} \)
29 \( 1 + (-1.17 + 3.61i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-0.690 + 0.502i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.927 + 2.85i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.951 - 2.92i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 1.62iT - 43T^{2} \)
47 \( 1 + (6.96 - 2.26i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.85 - 3.92i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (-2.48 - 0.809i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (2.5 - 3.44i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + 8.32T + 67T^{2} \)
71 \( 1 + (-6.06 + 8.35i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (14.4 + 4.70i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-6.28 - 8.64i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (-11.7 - 8.51i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 9.47iT - 89T^{2} \)
97 \( 1 + (-5.35 + 3.88i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.41431199280081424649148479627, −10.45329087346508335000978231406, −9.694712951745955250942957171260, −8.777639944257073760991319063124, −7.83324147258470131596984860799, −5.97194115760649828649417068043, −5.00596402827226760492594864492, −4.33692816221047858844447637049, −3.12298501473450541086909625309, −2.02504153201731218128099537840, 2.15695794138535898997767186750, 3.31182478484777482109723416886, 4.61508540997712200367672792801, 6.08580463822657725269627064478, 6.60697865676546294101231552863, 7.22666818541858779191447569579, 8.510150108323553953992926844078, 9.607650305622289500928437799782, 10.57751792706712548851631125025, 11.89734259537055625977149494394

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