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Properties

Label 2-363-33.2-c1-0-6
Degree $2$
Conductor $363$
Sign $-0.224 - 0.974i$
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 + (−1.29 − 1.15i)3-s + (−3.01 − 2.19i)4-s + (−1.17 + 0.382i)5-s + (3.58 − 2.08i)6-s + (1.66 − 2.28i)7-s + (3.35 − 2.43i)8-s + (0.334 + 2.98i)9-s − 2.96i·10-s + (1.36 + 6.31i)12-s + (3.18 + 1.03i)13-s + (3.98 + 5.47i)14-s + (1.96 + 0.866i)15-s + (0.761 + 2.34i)16-s + (1.28 + 3.94i)17-s + (−7.03 − 1.44i)18-s + ⋯
L(s)  = 1  + (−0.523 + 1.61i)2-s + (−0.745 − 0.666i)3-s + (−1.50 − 1.09i)4-s + (−0.527 + 0.171i)5-s + (1.46 − 0.851i)6-s + (0.628 − 0.864i)7-s + (1.18 − 0.861i)8-s + (0.111 + 0.993i)9-s − 0.938i·10-s + (0.394 + 1.82i)12-s + (0.882 + 0.286i)13-s + (1.06 + 1.46i)14-s + (0.507 + 0.223i)15-s + (0.190 + 0.585i)16-s + (0.310 + 0.956i)17-s + (−1.65 − 0.340i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.224 - 0.974i$
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: \((2,\ 363,\ (\ :1/2),\ -0.224 - 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.435993 + 0.548082i\)
\(L(\frac12)\) \(\approx\) \(0.435993 + 0.548082i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.29 + 1.15i)T \)
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

−11.49805665416088869441237587337, −10.80478492286227814023655842421, −9.706921506307231587240927383353, −8.098811830955031028217206224572, −8.043497010330145087453963153748, −6.93724412837446866530861670503, −6.25108708804108693911414358746, −5.23667304414840354761186269817, −4.07474775656441629002862161779, −1.15169988379174385142426893455, 0.822359618589084348815804395678, 2.64720183904582936016977089190, 3.89520552949118349969569052307, 4.84909896533061154513825248230, 6.04121273184947976641868396490, 7.85446460534485741942293370456, 8.842647412595957123552968467394, 9.515726617196437644739484885537, 10.45639916685252256349655178634, 11.36279971541886118601085161074

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