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Properties

Label 2-363-33.14-c2-0-29
Degree $2$
Conductor $363$
Sign $0.850 + 0.526i$
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  + (−1.69 − 2.32i)2-s + (2.68 + 1.33i)3-s + (−1.32 + 4.07i)4-s + (2.90 − 4.00i)5-s + (−1.42 − 8.51i)6-s + (−3.45 + 10.6i)7-s + (0.767 − 0.249i)8-s + (5.41 + 7.18i)9-s − 14.2·10-s + (−9.00 + 9.15i)12-s + (13.6 − 9.93i)13-s + (30.5 − 9.94i)14-s + (13.1 − 6.85i)15-s + (11.9 + 8.69i)16-s + (0.0367 − 0.0506i)17-s + (7.58 − 24.7i)18-s + ⋯
L(s)  = 1  + (−0.845 − 1.16i)2-s + (0.894 + 0.446i)3-s + (−0.330 + 1.01i)4-s + (0.581 − 0.800i)5-s + (−0.237 − 1.41i)6-s + (−0.493 + 1.51i)7-s + (0.0959 − 0.0311i)8-s + (0.601 + 0.798i)9-s − 1.42·10-s + (−0.750 + 0.763i)12-s + (1.05 − 0.764i)13-s + (2.18 − 0.710i)14-s + (0.877 − 0.456i)15-s + (0.748 + 0.543i)16-s + (0.00216 − 0.00297i)17-s + (0.421 − 1.37i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.850 + 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.850 + 0.526i$
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: \((2,\ 363,\ (\ :1),\ 0.850 + 0.526i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.50496 - 0.428065i\)
\(L(\frac12)\) \(\approx\) \(1.50496 - 0.428065i\)
\(L(2)\) 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 \( 1 + (1.69 + 2.32i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (-2.90 + 4.00i)T + (-7.72 - 23.7i)T^{2} \)
7 \( 1 + (3.45 - 10.6i)T + (-39.6 - 28.8i)T^{2} \)
13 \( 1 + (-13.6 + 9.93i)T + (52.2 - 160. i)T^{2} \)
17 \( 1 + (-0.0367 + 0.0506i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (-1.66 - 5.11i)T + (-292. + 212. i)T^{2} \)
23 \( 1 - 8.69iT - 529T^{2} \)
29 \( 1 + (-47.5 - 15.4i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-19.8 + 14.4i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (12.1 - 37.3i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-41.7 + 13.5i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 0.201T + 1.84e3T^{2} \)
47 \( 1 + (14.1 - 4.60i)T + (1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (57.5 + 79.2i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (-13.7 - 4.47i)T + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (23.7 + 17.2i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 - 82.8T + 4.48e3T^{2} \)
71 \( 1 + (15.6 - 21.6i)T + (-1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-4.31 + 13.2i)T + (-4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (40.6 - 29.4i)T + (1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (-35.4 + 48.7i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 40.3iT - 7.92e3T^{2} \)
97 \( 1 + (-24.9 + 18.1i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−10.90646904030963282717996241014, −9.907320367934598678931990787719, −9.425539497205139947937573648556, −8.567483224169696156859331818089, −8.269526164459859653842299159438, −6.14244276059160677454503178874, −5.11524711054123081273231899744, −3.40153388847204917182798361090, −2.55993920489539134089198503629, −1.36848248538966421861547187393, 0.996025374422668756997850998324, 2.90118133692238021654975936044, 4.14347675387404753473290359740, 6.37158829188719787724656365911, 6.59576051512297645062159665020, 7.46208931717846695850185385075, 8.330919525503257894768557172669, 9.277669557541861543941023120914, 10.05027989618291553509705126188, 10.81625819708825708010813153912

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