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Properties

Label 2-363-33.8-c1-0-15
Degree $2$
Conductor $363$
Sign $0.859 - 0.511i$
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  + (0.809 − 0.587i)2-s + (−1.03 + 1.38i)3-s + (−0.309 + 0.951i)4-s + (1.66 − 2.28i)5-s + (−0.0222 + 1.73i)6-s + (1.34 + 0.437i)7-s + (0.927 + 2.85i)8-s + (−0.853 − 2.87i)9-s − 2.82i·10-s + (−0.999 − 1.41i)12-s + (2.49 + 3.43i)13-s + (1.34 − 0.437i)14-s + (1.45 + 4.67i)15-s + (0.809 + 0.587i)16-s + (4.85 + 3.52i)17-s + (−2.38 − 1.82i)18-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.598 + 0.801i)3-s + (−0.154 + 0.475i)4-s + (0.743 − 1.02i)5-s + (−0.00907 + 0.707i)6-s + (0.508 + 0.165i)7-s + (0.327 + 1.00i)8-s + (−0.284 − 0.958i)9-s − 0.894i·10-s + (−0.288 − 0.408i)12-s + (0.691 + 0.951i)13-s + (0.359 − 0.116i)14-s + (0.375 + 1.20i)15-s + (0.202 + 0.146i)16-s + (1.17 + 0.855i)17-s + (−0.561 − 0.430i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.859 - 0.511i$
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: \((2,\ 363,\ (\ :1/2),\ 0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63716 + 0.450782i\)
\(L(\frac12)\) \(\approx\) \(1.63716 + 0.450782i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.03 - 1.38i)T \)
11 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.66 + 2.28i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (-1.34 - 0.437i)T + (5.66 + 4.11i)T^{2} \)
13 \( 1 + (-2.49 - 3.43i)T + (-4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.85 - 3.52i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.03 - 1.31i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-0.618 + 1.90i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.61 + 1.17i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.47 + 7.60i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (1.85 + 5.70i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 + (2.68 - 0.874i)T + (38.0 - 27.6i)T^{2} \)
53 \( 1 + (3.32 + 4.57i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (10.7 + 3.49i)T + (47.7 + 34.6i)T^{2} \)
61 \( 1 + (-5.81 + 8.00i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (-1.66 + 2.28i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (-1.34 - 0.437i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-2.49 - 3.43i)T + (-24.4 + 75.1i)T^{2} \)
83 \( 1 + (12.9 + 9.40i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (-1.61 + 1.17i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.57304819356842413011919480265, −10.81248773275877029287394195194, −9.697561367161279951017701834084, −8.831091741665054130464672337587, −8.093011321590100677632535047949, −6.23580191892110503309236153142, −5.36574690089499183468973654924, −4.51056512854637323092813105710, −3.66284185601900276675392624022, −1.77455971032798835276554107537, 1.27356201544770830159433021649, 2.94413981217429274234868851825, 4.77604864721591887311190040098, 5.71393014784949276217280695320, 6.37780899503190884322647699487, 7.19011402709413792976438102271, 8.247942898970579481874759967398, 9.858583488167890327257147143164, 10.53953608102139038057351855460, 11.23443941666547172034925266293

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