Properties

Label 2-363-33.5-c2-0-29
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: $\chi_{363} (269, \cdot )$
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

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

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