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Properties

Label 2-363-33.2-c1-0-3
Degree $2$
Conductor $363$
Sign $-0.591 + 0.806i$
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  + (−0.739 + 2.27i)2-s + (0.698 + 1.58i)3-s + (−3.01 − 2.19i)4-s + (1.17 − 0.382i)5-s + (−4.12 + 0.419i)6-s + (−1.66 + 2.28i)7-s + (3.35 − 2.43i)8-s + (−2.02 + 2.21i)9-s + 2.96i·10-s + (1.36 − 6.31i)12-s + (−3.18 − 1.03i)13-s + (−3.98 − 5.47i)14-s + (1.43 + 1.60i)15-s + (0.761 + 2.34i)16-s + (1.28 + 3.94i)17-s + (−3.54 − 6.24i)18-s + ⋯
L(s)  = 1  + (−0.523 + 1.61i)2-s + (0.403 + 0.914i)3-s + (−1.50 − 1.09i)4-s + (0.527 − 0.171i)5-s + (−1.68 + 0.171i)6-s + (−0.628 + 0.864i)7-s + (1.18 − 0.861i)8-s + (−0.674 + 0.738i)9-s + 0.938i·10-s + (0.394 − 1.82i)12-s + (−0.882 − 0.286i)13-s + (−1.06 − 1.46i)14-s + (0.369 + 0.413i)15-s + (0.190 + 0.585i)16-s + (0.310 + 0.956i)17-s + (−0.836 − 1.47i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.591 + 0.806i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.591 + 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.382689 - 0.755095i\)
\(L(\frac12)\) \(\approx\) \(0.382689 - 0.755095i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.698 - 1.58i)T \)
11 \( 1 \)
good2 \( 1 + (0.739 - 2.27i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (-1.17 + 0.382i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (1.66 - 2.28i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (3.18 + 1.03i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.28 - 3.94i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (1.43 + 1.98i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (-4.39 - 3.19i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.0606 + 0.186i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-8.46 - 6.15i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-5.81 + 4.22i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 4.24iT - 43T^{2} \)
47 \( 1 + (-1.99 - 2.73i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-6.75 - 2.19i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (7.42 - 10.2i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.29 - 1.39i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (7.93 - 2.57i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-4.76 + 6.55i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (6.98 + 2.27i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-4.58 - 14.1i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 9.58iT - 89T^{2} \)
97 \( 1 + (-0.700 + 2.15i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.12070463346231192990133192632, −10.55688704163088502706199094633, −9.719272706133805784594448435931, −9.138439859254164415318747910524, −8.415267076994932871566386315563, −7.43356545280951120021515629241, −6.11393310451614392078861324194, −5.55634913729085529335029307298, −4.48155178889040336306797343146, −2.72456687617754240152445500074, 0.64037252098584854826140065116, 2.14535441791001512762528915692, 3.01571785181561427467673411920, 4.26415542849605492039588409347, 6.15280873229556092137015577527, 7.26512152991845779657120938174, 8.199398653367615129118133672924, 9.543979304171718413295334126278, 9.729897732367120201716326293742, 10.83520316525170197576114467351

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