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Properties

Label 2-363-33.26-c2-0-5
Degree $2$
Conductor $363$
Sign $-0.786 - 0.617i$
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  + (−0.399 + 0.550i)2-s + (−2.18 − 2.05i)3-s + (1.09 + 3.36i)4-s + (3.81 + 5.25i)5-s + (2.00 − 0.377i)6-s + (2.89 + 8.90i)7-s + (−4.87 − 1.58i)8-s + (0.516 + 8.98i)9-s − 4.41·10-s + (4.54 − 9.58i)12-s + (3.37 + 2.45i)13-s + (−6.06 − 1.96i)14-s + (2.49 − 19.3i)15-s + (−8.62 + 6.26i)16-s + (−17.5 − 24.2i)17-s + (−5.15 − 3.30i)18-s + ⋯
L(s)  = 1  + (−0.199 + 0.275i)2-s + (−0.727 − 0.686i)3-s + (0.273 + 0.840i)4-s + (0.762 + 1.05i)5-s + (0.334 − 0.0628i)6-s + (0.413 + 1.27i)7-s + (−0.609 − 0.198i)8-s + (0.0573 + 0.998i)9-s − 0.441·10-s + (0.378 − 0.799i)12-s + (0.259 + 0.188i)13-s + (−0.432 − 0.140i)14-s + (0.166 − 1.28i)15-s + (−0.538 + 0.391i)16-s + (−1.03 − 1.42i)17-s + (−0.286 − 0.183i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.786 - 0.617i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ -0.786 - 0.617i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.383304 + 1.10966i\)
\(L(\frac12)\) \(\approx\) \(0.383304 + 1.10966i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.18 + 2.05i)T \)
11 \( 1 \)
good2 \( 1 + (0.399 - 0.550i)T + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (-3.81 - 5.25i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (-2.89 - 8.90i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (-3.37 - 2.45i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (17.5 + 24.2i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (-3.18 + 9.78i)T + (-292. - 212. i)T^{2} \)
23 \( 1 - 27.8iT - 529T^{2} \)
29 \( 1 + (15.3 - 4.99i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-14.2 - 10.3i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-4.49 - 13.8i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (-33.2 - 10.8i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 64.3T + 1.84e3T^{2} \)
47 \( 1 + (-9.94 - 3.23i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (35.5 - 48.8i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (-38.6 + 12.5i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (19.0 - 13.8i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 30.8T + 4.48e3T^{2} \)
71 \( 1 + (0.585 + 0.805i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (28.8 + 88.8i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (-91.4 - 66.4i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-61.1 - 84.1i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + 20.3iT - 7.92e3T^{2} \)
97 \( 1 + (153. + 111. i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−11.46233057699941866348809721797, −11.11472992114844213292047887961, −9.612890476670678110724576228691, −8.711804186508151620406808636516, −7.56402244937120388312179096834, −6.78672258434565405984424233269, −6.07901473972015433920819628394, −5.00192772600029911637596597376, −2.91488646925731451243088646607, −2.09861917273426912980696553771, 0.60970365084767744580024097600, 1.72809368520335117025954104045, 4.03444468154036530499867047592, 4.90237402749813917013009019684, 5.88132129753692646716845851069, 6.65780040634871719094641714701, 8.342428116827730007906261681990, 9.292754850747963609153515228538, 10.18394489875015889014652752675, 10.64790448747790045309542267961

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