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Properties

Label 2-363-33.14-c2-0-58
Degree $2$
Conductor $363$
Sign $-0.927 + 0.374i$
Analytic cond. $9.89103$
Root an. cond. $3.14500$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.927 − 2.85i)3-s + (1.23 − 3.80i)4-s + (3.39 − 10.4i)7-s + (−7.28 + 5.29i)9-s − 12·12-s + (17.7 − 12.9i)13-s + (−12.9 − 9.40i)16-s + (3.39 + 10.4i)19-s − 33.0·21-s + (7.72 + 23.7i)25-s + (21.8 + 15.8i)27-s + (−35.5 − 25.8i)28-s + (−47.7 + 34.6i)31-s + (11.1 + 34.2i)36-s + (14.5 − 44.6i)37-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (0.309 − 0.951i)4-s + (0.485 − 1.49i)7-s + (−0.809 + 0.587i)9-s − 12-s + (1.36 − 0.994i)13-s + (−0.809 − 0.587i)16-s + (0.178 + 0.550i)19-s − 1.57·21-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (−1.27 − 0.923i)28-s + (−1.53 + 1.11i)31-s + (0.309 + 0.951i)36-s + (0.392 − 1.20i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.927 + 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.927 + 0.374i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ -0.927 + 0.374i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.299171 - 1.54025i\)
\(L(\frac12)\) \(\approx\) \(0.299171 - 1.54025i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.927 + 2.85i)T \)
11 \( 1 \)
good2 \( 1 + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (-7.72 - 23.7i)T^{2} \)
7 \( 1 + (-3.39 + 10.4i)T + (-39.6 - 28.8i)T^{2} \)
13 \( 1 + (-17.7 + 12.9i)T + (52.2 - 160. i)T^{2} \)
17 \( 1 + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-3.39 - 10.4i)T + (-292. + 212. i)T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 + (680. + 494. i)T^{2} \)
31 \( 1 + (47.7 - 34.6i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (-14.5 + 44.6i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + 22T + 1.84e3T^{2} \)
47 \( 1 + (1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (-97.8 - 71.1i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 13T + 4.48e3T^{2} \)
71 \( 1 + (-1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-44.1 + 136. i)T + (-4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (8.89 - 6.46i)T + (1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (-136. + 99.3i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−10.84496664793531707486356925014, −10.32181764644385775115668302769, −8.841238628927278658813896607353, −7.68768005656089282704866775039, −7.04215505413552682882435349110, −5.98498654991854585725794062243, −5.15724470706697634201194423425, −3.55435017674364872447878444352, −1.63492743310266792375409588904, −0.77585254900084085293090646975, 2.27209543408246231629897607337, 3.52407644886895000911815997032, 4.58830466396822457784968864669, 5.76948764680050794339843448110, 6.67438569572213904702614407871, 8.284656934099966608105930892350, 8.754093905388428162072480180587, 9.596888465857811887015280229003, 11.14231619250331991749734176275, 11.41615437986171513372934391820

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