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Properties

Label 2-363-33.14-c2-0-62
Degree $2$
Conductor $363$
Sign $-0.897 - 0.441i$
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  + (−0.587 − 0.809i)2-s + (1.50 − 2.59i)3-s + (0.927 − 2.85i)4-s + (−3.88 + 5.35i)5-s + (−2.98 + 0.303i)6-s + (2.73 − 8.42i)7-s + (−6.65 + 2.16i)8-s + (−4.44 − 7.82i)9-s + 6.61·10-s + (−6 − 6.70i)12-s + (−10.7 + 7.77i)13-s + (−8.42 + 2.73i)14-s + (8.01 + 18.1i)15-s + (−4.04 − 2.93i)16-s + (−3.52 + 4.85i)17-s + (−3.71 + 8.19i)18-s + ⋯
L(s)  = 1  + (−0.293 − 0.404i)2-s + (0.502 − 0.864i)3-s + (0.231 − 0.713i)4-s + (−0.777 + 1.07i)5-s + (−0.497 + 0.0506i)6-s + (0.390 − 1.20i)7-s + (−0.832 + 0.270i)8-s + (−0.494 − 0.869i)9-s + 0.661·10-s + (−0.5 − 0.559i)12-s + (−0.823 + 0.598i)13-s + (−0.601 + 0.195i)14-s + (0.534 + 1.21i)15-s + (−0.252 − 0.183i)16-s + (−0.207 + 0.285i)17-s + (−0.206 + 0.455i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.897 - 0.441i$
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: \((2,\ 363,\ (\ :1),\ -0.897 - 0.441i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.183597 + 0.788784i\)
\(L(\frac12)\) \(\approx\) \(0.183597 + 0.788784i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.50 + 2.59i)T \)
11 \( 1 \)
good2 \( 1 + (0.587 + 0.809i)T + (-1.23 + 3.80i)T^{2} \)
5 \( 1 + (3.88 - 5.35i)T + (-7.72 - 23.7i)T^{2} \)
7 \( 1 + (-2.73 + 8.42i)T + (-39.6 - 28.8i)T^{2} \)
13 \( 1 + (10.7 - 7.77i)T + (52.2 - 160. i)T^{2} \)
17 \( 1 + (3.52 - 4.85i)T + (-89.3 - 274. i)T^{2} \)
19 \( 1 + (2.81 + 8.67i)T + (-292. + 212. i)T^{2} \)
23 \( 1 + 17.5iT - 529T^{2} \)
29 \( 1 + (25.1 + 8.15i)T + (680. + 494. i)T^{2} \)
31 \( 1 + (-5.01 + 3.64i)T + (296. - 913. i)T^{2} \)
37 \( 1 + (6.05 - 18.6i)T + (-1.10e3 - 804. i)T^{2} \)
41 \( 1 + (-5.32 + 1.72i)T + (1.35e3 - 988. i)T^{2} \)
43 \( 1 - 26.2T + 1.84e3T^{2} \)
47 \( 1 + (-16.8 + 5.47i)T + (1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (13.0 + 18.0i)T + (-868. + 2.67e3i)T^{2} \)
59 \( 1 + (-20.8 - 6.77i)T + (2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (75.5 + 54.8i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + 76.7T + 4.48e3T^{2} \)
71 \( 1 + (38.6 - 53.2i)T + (-1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (4.64 - 14.2i)T + (-4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (-95.5 + 69.3i)T + (1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (-65.3 + 89.9i)T + (-2.12e3 - 6.55e3i)T^{2} \)
89 \( 1 + 97.6iT - 7.92e3T^{2} \)
97 \( 1 + (-98.0 + 71.2i)T + (2.90e3 - 8.94e3i)T^{2} \)
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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

−10.83704168622827086780611284633, −9.943804008479464020091009572560, −8.829853501656210535895588459548, −7.58518392531875013393886580804, −7.08725914241424645489569864365, −6.22525558152013356782899854697, −4.41175519137631510980357046468, −3.08645633535326335199908581920, −1.91421865351865709675656275078, −0.35566638566095665706264477729, 2.45208600824561780185174962645, 3.68009565774097883106351302957, 4.81413135991137314271478719612, 5.71086163212973447210742608446, 7.55713616599222122368795207391, 8.077552260980376156900461645464, 8.951672352614622713960729711260, 9.364882469109258387252247023732, 10.86353753565547987896564768148, 11.97333982197949615052803283862

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