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Properties

Label 2-363-33.20-c2-0-11
Degree $2$
Conductor $363$
Sign $0.758 - 0.651i$
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  + (0.647 + 0.210i)2-s + (−2.63 − 1.43i)3-s + (−2.86 − 2.07i)4-s + (−6.17 + 2.00i)5-s + (−1.40 − 1.48i)6-s + (−7.57 − 5.50i)7-s + (−3.01 − 4.14i)8-s + (4.86 + 7.57i)9-s − 4.41·10-s + (4.54 + 9.58i)12-s + (−1.28 + 3.96i)13-s + (−3.74 − 5.15i)14-s + (19.1 + 3.59i)15-s + (3.29 + 10.1i)16-s + (28.4 − 9.24i)17-s + (1.55 + 5.92i)18-s + ⋯
L(s)  = 1  + (0.323 + 0.105i)2-s + (−0.877 − 0.479i)3-s + (−0.715 − 0.519i)4-s + (−1.23 + 0.401i)5-s + (−0.233 − 0.247i)6-s + (−1.08 − 0.786i)7-s + (−0.376 − 0.518i)8-s + (0.540 + 0.841i)9-s − 0.441·10-s + (0.378 + 0.799i)12-s + (−0.0991 + 0.305i)13-s + (−0.267 − 0.368i)14-s + (1.27 + 0.239i)15-s + (0.205 + 0.633i)16-s + (1.67 − 0.543i)17-s + (0.0864 + 0.329i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.758 - 0.651i$
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: \((2,\ 363,\ (\ :1),\ 0.758 - 0.651i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.375996 + 0.139202i\)
\(L(\frac12)\) \(\approx\) \(0.375996 + 0.139202i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.63 + 1.43i)T \)
11 \( 1 \)
good2 \( 1 + (-0.647 - 0.210i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (6.17 - 2.00i)T + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (7.57 + 5.50i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (1.28 - 3.96i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (-28.4 + 9.24i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (8.32 - 6.04i)T + (111. - 343. i)T^{2} \)
23 \( 1 + 27.8iT - 529T^{2} \)
29 \( 1 + (9.49 - 13.0i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (5.45 - 16.7i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (11.7 + 8.55i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-20.5 - 28.3i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 64.3T + 1.84e3T^{2} \)
47 \( 1 + (-6.14 - 8.45i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (-57.4 - 18.6i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-23.9 + 32.8i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-7.26 - 22.3i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 - 30.8T + 4.48e3T^{2} \)
71 \( 1 + (-0.946 + 0.307i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-75.6 - 54.9i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (34.9 - 107. i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (98.9 - 32.1i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 20.3iT - 7.92e3T^{2} \)
97 \( 1 + (-58.7 + 180. i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−11.38657009830753102663933961292, −10.37902395761433770932985777936, −9.836650370419069226526996458167, −8.327199328331250644019816595315, −7.22695432591808990867667266600, −6.60346690594698805652975734125, −5.46791829580554669512962585625, −4.31869085111991219946173374840, −3.41307436227605879021130986292, −0.800845676582910611864817453689, 0.29265355601282059714257359454, 3.34304141826390220577006668254, 3.91620597965183513878535560936, 5.15941745934543244008742661270, 5.91313708192402080472534860175, 7.38996908320774919101351894628, 8.367496859661545901334166901111, 9.347374377386210954885671881977, 10.11568299104000048492967332991, 11.50835972950149903682538482233

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