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Properties

Label 2-363-11.9-c1-0-11
Degree $2$
Conductor $363$
Sign $0.263 + 0.964i$
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  + (0.618 − 1.90i)2-s + (0.809 − 0.587i)3-s + (−1.61 − 1.17i)4-s + (1.23 + 3.80i)5-s + (−0.618 − 1.90i)6-s + (0.809 + 0.587i)7-s + (0.309 − 0.951i)9-s + 7.99·10-s − 1.99·12-s + (0.618 − 1.90i)13-s + (1.61 − 1.17i)14-s + (3.23 + 2.35i)15-s + (−1.23 − 3.80i)16-s + (−1.23 − 3.80i)17-s + (−1.61 − 1.17i)18-s + (−2.42 + 1.76i)19-s + ⋯
L(s)  = 1  + (0.437 − 1.34i)2-s + (0.467 − 0.339i)3-s + (−0.809 − 0.587i)4-s + (0.552 + 1.70i)5-s + (−0.252 − 0.776i)6-s + (0.305 + 0.222i)7-s + (0.103 − 0.317i)9-s + 2.52·10-s − 0.577·12-s + (0.171 − 0.527i)13-s + (0.432 − 0.314i)14-s + (0.835 + 0.607i)15-s + (−0.309 − 0.951i)16-s + (−0.299 − 0.922i)17-s + (−0.381 − 0.277i)18-s + (−0.556 + 0.404i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71846 - 1.31206i\)
\(L(\frac12)\) \(\approx\) \(1.71846 - 1.31206i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.618 + 1.90i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-1.23 - 3.80i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-0.618 + 1.90i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.23 + 3.80i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.42 - 1.76i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + (-4.85 - 3.52i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.54 - 4.75i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.42 + 1.76i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.61 - 1.17i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + (1.61 - 1.17i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.85 + 5.70i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.09 - 5.87i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.927 + 2.85i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + T + 67T^{2} \)
71 \( 1 + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (8.89 + 6.46i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.39 - 10.4i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.85 + 5.70i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + (-1.54 + 4.75i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.24868248685575492083375035486, −10.43355978283587790423559035483, −9.923985517587944492859195644037, −8.642366406757221136717410721457, −7.28162140044125638184862420894, −6.56529712998703429070053798983, −5.10108224300748058215309484366, −3.48844896291684355147802204782, −2.82630080265690501179832189184, −1.83813456029861339633822950396, 1.77342867087038813933050586413, 4.19009347778286151039189145019, 4.76120497561502996147941142752, 5.72102850617594003816902025012, 6.72051016256054584965644855639, 8.095960379650779406713820007232, 8.535296350530508220496109766154, 9.361857442229668437196602957612, 10.51114898130113743638293183219, 11.82786304164387173035538290214

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