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Properties

Label 2-363-11.3-c1-0-2
Degree $2$
Conductor $363$
Sign $-0.659 - 0.751i$
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  + (−1.40 + 1.01i)2-s + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (2.42 + 1.76i)5-s + (1.40 + 1.01i)6-s + (−1.07 + 3.29i)7-s + (−0.535 − 1.64i)8-s + (−0.809 + 0.587i)9-s − 5.19·10-s − 0.999·12-s + (1.40 − 1.01i)13-s + (−1.85 − 5.70i)14-s + (0.927 − 2.85i)15-s + (4.04 + 2.93i)16-s + (−1.40 − 1.01i)17-s + (0.535 − 1.64i)18-s + ⋯
L(s)  = 1  + (−0.990 + 0.719i)2-s + (−0.178 − 0.549i)3-s + (0.154 − 0.475i)4-s + (1.08 + 0.788i)5-s + (0.572 + 0.415i)6-s + (−0.404 + 1.24i)7-s + (−0.189 − 0.582i)8-s + (−0.269 + 0.195i)9-s − 1.64·10-s − 0.288·12-s + (0.388 − 0.282i)13-s + (−0.495 − 1.52i)14-s + (0.239 − 0.736i)15-s + (1.01 + 0.734i)16-s + (−0.339 − 0.246i)17-s + (0.126 − 0.388i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.659 - 0.751i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.296269 + 0.653897i\)
\(L(\frac12)\) \(\approx\) \(0.296269 + 0.653897i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.309 + 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (1.40 - 1.01i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-2.42 - 1.76i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (1.07 - 3.29i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.40 + 1.01i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.40 + 1.01i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.14 - 6.58i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (0.535 - 1.64i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.23 - 2.35i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (3.39 - 10.4i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.535 + 1.64i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.46T + 43T^{2} \)
47 \( 1 + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-7.28 + 5.29i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.85 - 5.70i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 2T + 67T^{2} \)
71 \( 1 + (-4.85 - 3.52i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.14 + 6.58i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 9T + 89T^{2} \)
97 \( 1 + (-5.66 + 4.11i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.84261289678431140258819798826, −10.40819516355744906395345353561, −9.825918203469499321583799647888, −8.887424843880581459253936569320, −8.085211006446370885746700496722, −6.99004298891475597601711863120, −6.11371113063223955329067002202, −5.68636494407671867570284717921, −3.26128540389374304852470714138, −1.88294261858518049186058766262, 0.69066536999235802947591352818, 2.14891460825547206488438341386, 3.88575524645190048200182066540, 5.13477699623572997110190246485, 6.17968794026763141748679282076, 7.55306187285961558761778222401, 8.922216903820059997046422527985, 9.327205882649217624252019710871, 10.14903837265714745238669008090, 10.75760871201764765517674130351

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