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Properties

Label 2-363-33.17-c1-0-24
Degree $2$
Conductor $363$
Sign $-0.640 + 0.767i$
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  + (0.570 − 1.63i)3-s + (1.61 − 1.17i)4-s + (−3.15 − 1.02i)5-s + (−2.34 − 1.86i)9-s + (−0.999 − 3.31i)12-s + (−3.47 + 4.57i)15-s + (1.23 − 3.80i)16-s + (−6.30 + 2.04i)20-s − 3.31i·23-s + (4.85 + 3.52i)25-s + (−4.39 + 2.77i)27-s + (1.54 + 4.75i)31-s + (−5.99 − 0.255i)36-s + (5.66 − 4.11i)37-s + (5.49 + 8.29i)45-s + ⋯
L(s)  = 1  + (0.329 − 0.944i)3-s + (0.809 − 0.587i)4-s + (−1.41 − 0.458i)5-s + (−0.783 − 0.621i)9-s + (−0.288 − 0.957i)12-s + (−0.897 + 1.18i)15-s + (0.309 − 0.951i)16-s + (−1.41 + 0.458i)20-s − 0.691i·23-s + (0.970 + 0.705i)25-s + (−0.844 + 0.534i)27-s + (0.277 + 0.854i)31-s + (−0.999 − 0.0426i)36-s + (0.931 − 0.676i)37-s + (0.819 + 1.23i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.640 + 0.767i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.640 + 0.767i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.530696 - 1.13402i\)
\(L(\frac12)\) \(\approx\) \(0.530696 - 1.13402i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.570 + 1.63i)T \)
11 \( 1 \)
good2 \( 1 + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (3.15 + 1.02i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (-2.16 + 6.65i)T^{2} \)
13 \( 1 + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.31iT - 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-1.54 - 4.75i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-5.66 + 4.11i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-3.89 + 5.36i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-12.6 + 4.09i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.94 - 2.68i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (49.3 + 35.8i)T^{2} \)
67 \( 1 + 13T + 67T^{2} \)
71 \( 1 + (-15.7 - 5.12i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-22.5 + 69.4i)T^{2} \)
79 \( 1 + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 + (-5.25 - 16.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.38724532908201489301473884083, −10.37866245717594095544940576717, −8.974551064425150980543415204519, −8.123068013780067704962466187735, −7.31730477515181662426902423026, −6.55940674369562855278240582206, −5.31018345870033143730483222875, −3.81703828548082123110047151839, −2.44952051215274306660171749795, −0.824933596876542674844856619446, 2.68348368065515614122209828544, 3.60821845273483581345198959887, 4.42517317510318005073199156116, 6.02878276290360590832050377772, 7.37969801341615230240998558800, 7.891298605044527616161070124593, 8.855244847220158548153243612721, 10.09392899310729892948592392296, 11.07448548870139015605254034627, 11.49341028834857870363521876118

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