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Properties

Label 2-363-33.29-c1-0-7
Degree $2$
Conductor $363$
Sign $-0.352 - 0.935i$
Analytic cond. $2.89856$
Root an. cond. $1.70251$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.21 + 0.885i)2-s + (0.590 + 1.62i)3-s + (0.0828 + 0.254i)4-s + (1.71 + 2.35i)5-s + (−0.721 + 2.50i)6-s + (−2.68 + 0.874i)7-s + (0.806 − 2.48i)8-s + (−2.30 + 1.92i)9-s + 4.38i·10-s + (−0.366 + 0.285i)12-s + (0.526 − 0.725i)13-s + (−4.05 − 1.31i)14-s + (−2.82 + 4.17i)15-s + (3.61 − 2.62i)16-s + (2.11 − 1.53i)17-s + (−4.50 + 0.305i)18-s + ⋯
L(s)  = 1  + (0.861 + 0.625i)2-s + (0.341 + 0.940i)3-s + (0.0414 + 0.127i)4-s + (0.764 + 1.05i)5-s + (−0.294 + 1.02i)6-s + (−1.01 + 0.330i)7-s + (0.284 − 0.877i)8-s + (−0.767 + 0.641i)9-s + 1.38i·10-s + (−0.105 + 0.0823i)12-s + (0.146 − 0.201i)13-s + (−1.08 − 0.351i)14-s + (−0.728 + 1.07i)15-s + (0.902 − 0.655i)16-s + (0.511 − 0.371i)17-s + (−1.06 + 0.0719i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.352 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $-0.352 - 0.935i$
Analytic conductor: \(2.89856\)
Root analytic conductor: \(1.70251\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{363} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 363,\ (\ :1/2),\ -0.352 - 0.935i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29485 + 1.87211i\)
\(L(\frac12)\) \(\approx\) \(1.29485 + 1.87211i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.590 - 1.62i)T \)
11 \( 1 \)
good2 \( 1 + (-1.21 - 0.885i)T + (0.618 + 1.90i)T^{2} \)
5 \( 1 + (-1.71 - 2.35i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (2.68 - 0.874i)T + (5.66 - 4.11i)T^{2} \)
13 \( 1 + (-0.526 + 0.725i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.11 + 1.53i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-2.32 - 0.756i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (2.66 + 8.20i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-8.24 - 5.99i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.09 + 3.36i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.39 - 4.29i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.24iT - 43T^{2} \)
47 \( 1 + (-2.02 - 0.658i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.87 + 5.33i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.542 - 0.176i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (5.98 + 8.23i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 + (5.58 + 7.69i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-3.05 + 0.991i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.31 - 5.94i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-5.11 + 3.71i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 + (4.63 + 3.36i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.71935331094482821406325660295, −10.41956950884657729040216066204, −9.952621961719944467110059381398, −9.285706653754103677205165824288, −7.75436755266716850089978300191, −6.48701499184256075000357922960, −5.95309333017009649052780029431, −4.93317103657952528251773041985, −3.58580749273460449560051020835, −2.76666886963905716799163156624, 1.36139087249496347041035081767, 2.73857826280992008119888291090, 3.82236525004929038334152491086, 5.22059019526324723391055235461, 6.09958813415160630262331670093, 7.30141238884934937641330106539, 8.460814327783749308563611848379, 9.220165443274005699295010458438, 10.27904313234624981040991215839, 11.61948460954462553857248026415

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