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Properties

Label 2-363-33.5-c2-0-45
Degree $2$
Conductor $363$
Sign $0.977 - 0.212i$
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 + (−0.440 + 2.96i)3-s + (6.67 − 4.85i)4-s + (7.51 + 2.44i)5-s + (1.74 + 10.3i)6-s + (−1.46 + 1.06i)7-s + (8.75 − 12.0i)8-s + (−8.61 − 2.61i)9-s + 27.6·10-s + (11.4 + 21.9i)12-s + (0.0219 + 0.0674i)13-s + (−3.73 + 5.13i)14-s + (−10.5 + 21.2i)15-s + (5.91 − 18.2i)16-s + (3.48 + 1.13i)17-s + (−31.5 + 0.612i)18-s + ⋯
L(s)  = 1  + (1.66 − 0.540i)2-s + (−0.146 + 0.989i)3-s + (1.66 − 1.21i)4-s + (1.50 + 0.488i)5-s + (0.290 + 1.72i)6-s + (−0.209 + 0.152i)7-s + (1.09 − 1.50i)8-s + (−0.956 − 0.290i)9-s + 2.76·10-s + (0.954 + 1.82i)12-s + (0.00168 + 0.00518i)13-s + (−0.266 + 0.367i)14-s + (−0.703 + 1.41i)15-s + (0.369 − 1.13i)16-s + (0.204 + 0.0665i)17-s + (−1.75 + 0.0340i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.977 - 0.212i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.977 - 0.212i$
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: \((2,\ 363,\ (\ :1),\ 0.977 - 0.212i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.77968 + 0.514218i\)
\(L(\frac12)\) \(\approx\) \(4.77968 + 0.514218i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.440 - 2.96i)T \)
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

−11.14240389305967423280889006832, −10.58897543303965566260216250828, −9.844196102609972166112343416342, −8.849059308848790789117685511762, −6.70103012808634496162270048657, −5.98999478141532484262909235681, −5.25593981564097782818740144237, −4.25673761249161194350418880207, −3.03317832630518320697138038960, −2.16324443172293603590303059359, 1.70425933992503874991617067677, 2.83940236372316820850097810437, 4.42923908771293132704405715234, 5.70731860118032261133186299928, 5.96214856819203324434966119549, 6.91605090384870321680778116115, 7.965821045633918487506115521360, 9.229308041384419056168638120153, 10.50619201551126389432702401434, 11.70429556315982732633867613457

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