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Properties

Label 2-363-33.14-c2-0-47
Degree $2$
Conductor $363$
Sign $0.917 - 0.398i$
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  + (1.55 + 2.14i)2-s + (2.38 − 1.82i)3-s + (−0.927 + 2.85i)4-s + (−1.66 + 2.28i)5-s + (7.60 + 2.25i)6-s + (2.31 − 7.11i)7-s + (2.51 − 0.817i)8-s + (2.33 − 8.69i)9-s − 7.48·10-s + (2.99 + 8.48i)12-s + (18.1 − 13.1i)13-s + (18.8 − 6.11i)14-s + (0.217 + 8.48i)15-s + (15.3 + 11.1i)16-s + (−12.4 + 17.1i)17-s + (22.2 − 8.51i)18-s + ⋯
L(s)  = 1  + (0.777 + 1.07i)2-s + (0.793 − 0.608i)3-s + (−0.231 + 0.713i)4-s + (−0.332 + 0.457i)5-s + (1.26 + 0.376i)6-s + (0.330 − 1.01i)7-s + (0.314 − 0.102i)8-s + (0.259 − 0.965i)9-s − 0.748·10-s + (0.249 + 0.707i)12-s + (1.39 − 1.01i)13-s + (1.34 − 0.437i)14-s + (0.0145 + 0.565i)15-s + (0.960 + 0.697i)16-s + (−0.731 + 1.00i)17-s + (1.23 − 0.472i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.917 - 0.398i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.917 - 0.398i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.34593 + 0.694591i\)
\(L(\frac12)\) \(\approx\) \(3.34593 + 0.694591i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.38 + 1.82i)T \)
11 \( 1 \)
good2 \( 1 + (-1.55 - 2.14i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (1.66 - 2.28i)T + (-7.72 - 23.7i)T^{2} \)
7 \( 1 + (-2.31 + 7.11i)T + (-39.6 - 28.8i)T^{2} \)
13 \( 1 + (-18.1 + 13.1i)T + (52.2 - 160. i)T^{2} \)
17 \( 1 + (12.4 - 17.1i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (4.62 + 14.2i)T + (-292. + 212. i)T^{2} \)
23 \( 1 - 31.1iT - 529T^{2} \)
29 \( 1 + (20.1 + 6.54i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (24.2 - 17.6i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (3.09 - 9.51i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-40.2 + 13.0i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 14.9T + 1.84e3T^{2} \)
47 \( 1 + (-34.9 + 11.3i)T + (1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (24.9 + 34.3i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (32.2 + 10.4i)T + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (78.7 + 57.1i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 42T + 4.48e3T^{2} \)
71 \( 1 + (38.2 - 52.6i)T + (-1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (23.1 - 71.1i)T + (-4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (-18.1 + 13.1i)T + (1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (12.4 - 17.1i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 62.2iT - 7.92e3T^{2} \)
97 \( 1 + (59.8 - 43.4i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−11.12561965448210741796671142041, −10.59931327297450067445824068891, −9.049166931069390755648070216012, −7.975454027027746917532521599306, −7.42695460956623528131441972123, −6.62073262731305323494774313015, −5.65192710887742025045582132211, −4.09919625820517563134672758252, −3.43933774121234987114915001757, −1.37999127503482149318345713318, 1.82146005612344603100205060608, 2.80427647259149426833008022753, 4.10134330269141871175898437530, 4.60166310431899429505184192890, 5.91792260499906788711979164540, 7.65478776270966473292203368551, 8.761359107377842652364096512502, 9.153279101696372222343362303023, 10.58026742094662203989512809350, 11.20651851640246457684504109933

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