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Properties

Label 2-363-33.14-c2-0-51
Degree $2$
Conductor $363$
Sign $0.762 - 0.646i$
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  + (2.05 + 2.83i)2-s + (2.10 − 2.14i)3-s + (−2.55 + 7.85i)4-s + (4.64 − 6.39i)5-s + (10.3 + 1.54i)6-s + (0.560 − 1.72i)7-s + (−14.1 + 4.60i)8-s + (−0.174 − 8.99i)9-s + 27.6·10-s + (11.4 + 21.9i)12-s + (−0.0573 + 0.0416i)13-s + (6.04 − 1.96i)14-s + (−3.93 − 23.3i)15-s + (−15.4 − 11.2i)16-s + (2.15 − 2.96i)17-s + (25.1 − 19.0i)18-s + ⋯
L(s)  = 1  + (1.02 + 1.41i)2-s + (0.700 − 0.713i)3-s + (−0.637 + 1.96i)4-s + (0.928 − 1.27i)5-s + (1.73 + 0.257i)6-s + (0.0801 − 0.246i)7-s + (−1.77 + 0.575i)8-s + (−0.0194 − 0.999i)9-s + 2.76·10-s + (0.954 + 1.82i)12-s + (−0.00441 + 0.00320i)13-s + (0.431 − 0.140i)14-s + (−0.262 − 1.55i)15-s + (−0.967 − 0.703i)16-s + (0.126 − 0.174i)17-s + (1.39 − 1.05i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.762 - 0.646i$
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: \((2,\ 363,\ (\ :1),\ 0.762 - 0.646i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.71549 + 1.36383i\)
\(L(\frac12)\) \(\approx\) \(3.71549 + 1.36383i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-2.10 + 2.14i)T \)
11 \( 1 \)
good2 \( 1 + (-2.05 - 2.83i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (-4.64 + 6.39i)T + (-7.72 - 23.7i)T^{2} \)
7 \( 1 + (-0.560 + 1.72i)T + (-39.6 - 28.8i)T^{2} \)
13 \( 1 + (0.0573 - 0.0416i)T + (52.2 - 160. i)T^{2} \)
17 \( 1 + (-2.15 + 2.96i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-8.13 - 25.0i)T + (-292. + 212. i)T^{2} \)
23 \( 1 + 6.84iT - 529T^{2} \)
29 \( 1 + (-28.7 - 9.33i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (34.1 - 24.8i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (-1.14 + 3.51i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (67.5 - 21.9i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 + 13.8T + 1.84e3T^{2} \)
47 \( 1 + (26.0 - 8.45i)T + (1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (-12.7 - 17.5i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (-75.8 - 24.6i)T + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (64.0 + 46.5i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 101.T + 4.48e3T^{2} \)
71 \( 1 + (-27.2 + 37.5i)T + (-1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-35.6 + 109. i)T + (-4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (50.8 - 36.9i)T + (1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (49.6 - 68.3i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 - 45.3iT - 7.92e3T^{2} \)
97 \( 1 + (-77.8 + 56.5i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−12.11998898128076128240672214469, −10.05647536061294759548757093166, −8.935558319028999816765638116776, −8.317661376648438666882227190477, −7.42324729471178614252381716639, −6.40995030595791619533883942464, −5.57343394073781826563671220257, −4.65488567131214737634989302217, −3.37646270448047828507770332281, −1.51289922648276216398881383753, 2.00213325113696690413773658965, 2.78871267853175424133299573641, 3.60930572023397813244285792485, 4.88638236031433891888811099529, 5.79710451506166628759621598954, 7.13449488199522376364154415913, 8.814158386522723280124046413381, 9.841497493168938819486946023083, 10.25458594220145346196337222060, 11.09419633375483543317199500655

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