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Properties

Label 2-363-33.20-c2-0-34
Degree $2$
Conductor $363$
Sign $-0.733 + 0.679i$
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  + (−3.32 − 1.08i)2-s + (2.68 + 1.33i)3-s + (6.67 + 4.85i)4-s + (−7.51 + 2.44i)5-s + (−7.49 − 7.35i)6-s + (−1.46 − 1.06i)7-s + (−8.75 − 12.0i)8-s + (5.43 + 7.17i)9-s + 27.6·10-s + (11.4 + 21.9i)12-s + (0.0219 − 0.0674i)13-s + (3.73 + 5.13i)14-s + (−23.4 − 3.47i)15-s + (5.91 + 18.2i)16-s + (−3.48 + 1.13i)17-s + (−10.3 − 29.7i)18-s + ⋯
L(s)  = 1  + (−1.66 − 0.540i)2-s + (0.895 + 0.445i)3-s + (1.66 + 1.21i)4-s + (−1.50 + 0.488i)5-s + (−1.24 − 1.22i)6-s + (−0.209 − 0.152i)7-s + (−1.09 − 1.50i)8-s + (0.603 + 0.797i)9-s + 2.76·10-s + (0.954 + 1.82i)12-s + (0.00168 − 0.00518i)13-s + (0.266 + 0.367i)14-s + (−1.56 − 0.231i)15-s + (0.369 + 1.13i)16-s + (−0.204 + 0.0665i)17-s + (−0.573 − 1.65i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.733 + 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.733 + 0.679i$
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: \((2,\ 363,\ (\ :1),\ -0.733 + 0.679i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0607345 - 0.154847i\)
\(L(\frac12)\) \(\approx\) \(0.0607345 - 0.154847i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.68 - 1.33i)T \)
11 \( 1 \)
good2 \( 1 + (3.32 + 1.08i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (7.51 - 2.44i)T + (20.2 - 14.6i)T^{2} \)
7 \( 1 + (1.46 + 1.06i)T + (15.1 + 46.6i)T^{2} \)
13 \( 1 + (-0.0219 + 0.0674i)T + (-136. - 99.3i)T^{2} \)
17 \( 1 + (3.48 - 1.13i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (21.2 - 15.4i)T + (111. - 343. i)T^{2} \)
23 \( 1 + 6.84iT - 529T^{2} \)
29 \( 1 + (-17.7 + 24.4i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (-13.0 + 40.1i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (2.99 + 2.17i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (41.7 + 57.5i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + 13.8T + 1.84e3T^{2} \)
47 \( 1 + (16.0 + 22.1i)T + (-682. + 2.10e3i)T^{2} \)
53 \( 1 + (20.6 + 6.71i)T + (2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-46.9 + 64.5i)T + (-1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-24.4 - 75.2i)T + (-3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 101.T + 4.48e3T^{2} \)
71 \( 1 + (44.1 - 14.3i)T + (4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (93.4 + 67.8i)T + (1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (-19.4 + 59.8i)T + (-5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (-80.3 + 26.1i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 - 45.3iT - 7.92e3T^{2} \)
97 \( 1 + (29.7 - 91.5i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−10.49526828974627513033626813483, −10.08506622738561192464672652951, −8.864035929096005642443174121546, −8.232431711119046184542807991529, −7.64508022504882662215666157655, −6.70865567393573086401745467213, −4.25068921276908026994840431571, −3.35339104022159008244639951863, −2.16253571408479942179058058744, −0.12362187009059141264472471499, 1.30154771899477567350183442947, 3.01455304644368109215555352314, 4.52749823927006433025883548778, 6.54421264190576829067027174614, 7.17551107009377243242292817778, 8.141706133193207351567420089990, 8.551086572926994130978618709199, 9.255687110727905646510542486979, 10.37371692750714012172634158236, 11.37605850951653001324898004780

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