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Properties

Label 2-363-33.20-c2-0-45
Degree $2$
Conductor $363$
Sign $0.263 + 0.964i$
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  + (2.42 − 1.76i)3-s + (−3.23 − 2.35i)4-s + (8.89 + 6.46i)7-s + (2.78 − 8.55i)9-s − 12·12-s + (6.79 − 20.9i)13-s + (4.94 + 15.2i)16-s + (8.89 − 6.46i)19-s + 33·21-s + (−20.2 + 14.6i)25-s + (−8.34 − 25.6i)27-s + (−13.5 − 41.8i)28-s + (18.2 − 56.1i)31-s + (−29.1 + 21.1i)36-s + (−38.0 − 27.6i)37-s + ⋯
L(s)  = 1  + (0.809 − 0.587i)3-s + (−0.809 − 0.587i)4-s + (1.27 + 0.923i)7-s + (0.309 − 0.951i)9-s − 12-s + (0.522 − 1.60i)13-s + (0.309 + 0.951i)16-s + (0.468 − 0.340i)19-s + 1.57·21-s + (−0.809 + 0.587i)25-s + (−0.309 − 0.951i)27-s + (−0.485 − 1.49i)28-s + (0.588 − 1.81i)31-s + (−0.809 + 0.587i)36-s + (−1.02 − 0.746i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.263 + 0.964i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.65146 - 1.26090i\)
\(L(\frac12)\) \(\approx\) \(1.65146 - 1.26090i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.42 + 1.76i)T \)
11 \( 1 \)
good2 \( 1 + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (-8.89 - 6.46i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (-6.79 + 20.9i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (233. - 169. i)T^{2} \)
19 \( 1 + (-8.89 + 6.46i)T + (111. - 343. i)T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 + (-259. - 799. i)T^{2} \)
31 \( 1 + (-18.2 + 56.1i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (38.0 + 27.6i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 22T + 1.84e3T^{2} \)
47 \( 1 + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-37.3 - 115. i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 13T + 4.48e3T^{2} \)
71 \( 1 + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-115. - 84.0i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (3.39 - 10.4i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 7.92e3T^{2} \)
97 \( 1 + (52.2 - 160. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−11.03803484608024622608205260189, −9.900985338065187170225606175024, −8.989132862580965105328090633291, −8.269011070139240004909007471010, −7.62535905706536169615552341177, −5.93066785938027438798963794168, −5.25571288478607986485766226376, −3.85854626850095864416095372911, −2.35458388331983318426914553385, −0.999783355947410086190317290308, 1.66593026482930875761614946093, 3.48638145136014149434363084150, 4.33016000537224231018635651305, 4.98537508723023379003924166893, 6.95716202983744500082611415111, 7.977995075790014923488877862851, 8.534070044487507148869919041100, 9.432063248828930706238616312948, 10.36657011091936243082861634794, 11.33041322214187061405569052999

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