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Properties

Label 2-363-11.2-c2-0-5
Degree $2$
Conductor $363$
Sign $0.416 - 0.909i$
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  + (0.297 + 0.0967i)2-s + (−1.40 + 1.01i)3-s + (−3.15 − 2.29i)4-s + (−2.29 − 7.05i)5-s + (−0.515 + 0.167i)6-s + (−5.89 + 8.11i)7-s + (−1.45 − 2.00i)8-s + (0.927 − 2.85i)9-s − 2.32i·10-s + 6.75·12-s + (−3.23 − 1.05i)13-s + (−2.53 + 1.84i)14-s + (10.3 + 7.54i)15-s + (4.58 + 14.1i)16-s + (14.7 − 4.80i)17-s + (0.552 − 0.759i)18-s + ⋯
L(s)  = 1  + (0.148 + 0.0483i)2-s + (−0.467 + 0.339i)3-s + (−0.789 − 0.573i)4-s + (−0.458 − 1.41i)5-s + (−0.0859 + 0.0279i)6-s + (−0.841 + 1.15i)7-s + (−0.181 − 0.250i)8-s + (0.103 − 0.317i)9-s − 0.232i·10-s + 0.563·12-s + (−0.249 − 0.0809i)13-s + (−0.181 + 0.131i)14-s + (0.692 + 0.503i)15-s + (0.286 + 0.881i)16-s + (0.870 − 0.282i)17-s + (0.0306 − 0.0422i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.416 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.416 - 0.909i$
Analytic conductor: \(9.89103\)
Root analytic conductor: \(3.14500\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1),\ 0.416 - 0.909i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.558365 + 0.358424i\)
\(L(\frac12)\) \(\approx\) \(0.558365 + 0.358424i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.40 - 1.01i)T \)
11 \( 1 \)
good2 \( 1 + (-0.297 - 0.0967i)T + (3.23 + 2.35i)T^{2} \)
5 \( 1 + (2.29 + 7.05i)T + (-20.2 + 14.6i)T^{2} \)
7 \( 1 + (5.89 - 8.11i)T + (-15.1 - 46.6i)T^{2} \)
13 \( 1 + (3.23 + 1.05i)T + (136. + 99.3i)T^{2} \)
17 \( 1 + (-14.7 + 4.80i)T + (233. - 169. i)T^{2} \)
19 \( 1 + (-18.0 - 24.7i)T + (-111. + 343. i)T^{2} \)
23 \( 1 - 7.67T + 529T^{2} \)
29 \( 1 + (-1.98 + 2.73i)T + (-259. - 799. i)T^{2} \)
31 \( 1 + (1.24 - 3.84i)T + (-777. - 564. i)T^{2} \)
37 \( 1 + (-1.92 - 1.39i)T + (423. + 1.30e3i)T^{2} \)
41 \( 1 + (-4.13 - 5.69i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 - 3.99iT - 1.84e3T^{2} \)
47 \( 1 + (39.9 - 29.0i)T + (682. - 2.10e3i)T^{2} \)
53 \( 1 + (18.4 - 56.8i)T + (-2.27e3 - 1.65e3i)T^{2} \)
59 \( 1 + (-8.89 - 6.46i)T + (1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (70.6 - 22.9i)T + (3.01e3 - 2.18e3i)T^{2} \)
67 \( 1 + 3.22T + 4.48e3T^{2} \)
71 \( 1 + (-36.1 - 111. i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-11.0 + 15.1i)T + (-1.64e3 - 5.06e3i)T^{2} \)
79 \( 1 + (-3.34 - 1.08i)T + (5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-139. + 45.4i)T + (5.57e3 - 4.04e3i)T^{2} \)
89 \( 1 + 65.8T + 7.92e3T^{2} \)
97 \( 1 + (-20.0 + 61.6i)T + (-7.61e3 - 5.53e3i)T^{2} \)
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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

−11.67236465338284299446256627288, −10.14856347694265461284648257405, −9.475252047940761072894090517219, −8.880616034497250426697402209672, −7.84986751456374347583457241857, −6.06473539339628785231635246539, −5.43710108185961590064071790817, −4.66833620444175474778415687385, −3.42199235648695028316658215543, −1.08127123452620849738375433090, 0.38875656862345734185941325614, 3.02469897863479398380393086932, 3.70035870604021285438679416276, 5.00833489916013122628180997890, 6.51515208706922442197260816534, 7.22464203991972495358224706586, 7.85836609310886411348898674555, 9.403587350139357967941829464851, 10.23184315993308407958209780791, 11.08486430368496370459750289155

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