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Properties

Label 2-363-11.5-c1-0-1
Degree $2$
Conductor $363$
Sign $0.649 - 0.760i$
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  + (−0.535 − 1.64i)2-s + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.927 + 2.85i)5-s + (0.535 − 1.64i)6-s + (−2.80 + 2.03i)7-s + (−1.40 − 1.01i)8-s + (0.309 + 0.951i)9-s + 5.19·10-s − 0.999·12-s + (0.535 + 1.64i)13-s + (4.85 + 3.52i)14-s + (−2.42 + 1.76i)15-s + (−1.54 + 4.75i)16-s + (−0.535 + 1.64i)17-s + (1.40 − 1.01i)18-s + ⋯
L(s)  = 1  + (−0.378 − 1.16i)2-s + (0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.414 + 1.27i)5-s + (0.218 − 0.672i)6-s + (−1.05 + 0.769i)7-s + (−0.495 − 0.359i)8-s + (0.103 + 0.317i)9-s + 1.64·10-s − 0.288·12-s + (0.148 + 0.456i)13-s + (1.29 + 0.942i)14-s + (−0.626 + 0.455i)15-s + (−0.386 + 1.18i)16-s + (−0.129 + 0.399i)17-s + (0.330 − 0.239i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.649 - 0.760i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (148, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.649 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.785999 + 0.362388i\)
\(L(\frac12)\) \(\approx\) \(0.785999 + 0.362388i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.535 + 1.64i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.927 - 2.85i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.80 - 2.03i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-0.535 - 1.64i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.535 - 1.64i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.60 - 4.07i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (1.40 - 1.01i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.23 - 3.80i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-8.89 + 6.46i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.40 + 1.01i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.46T + 43T^{2} \)
47 \( 1 + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.78 + 8.55i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-4.85 + 3.52i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + (1.85 - 5.70i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-5.60 + 4.07i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (2.16 + 6.65i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.47905423995959877401828610849, −10.54320973691470107944475690088, −9.853672491457069637944201354575, −9.246185363729126057189863226366, −8.036371051793658496396185998416, −6.78090816121610262672348377277, −5.89679214434912135120136067823, −3.76755041698659470355850878710, −3.19077166505048681194876320322, −2.15701784275697087471445720558, 0.61468849746258583656114674166, 3.04132081423641568611324830354, 4.44561055528657709223098012771, 5.71944854308034328836019189644, 6.74791401674020219433385531035, 7.64712807427755653644114459201, 8.235241101693750259159695153191, 9.263619356648617509955314645459, 9.819673198404563108401798520579, 11.51593123766139099721734716011

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