↑

Properties

Label 2-363-33.14-c2-0-53
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

Related objects

Downloads

Learn more

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: $\chi_{363} (245, \cdot )$
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 \( 1 + (153. - 111. i)T + (2.90e3 - 8.94e3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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