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Properties

Label 2-363-33.26-c2-0-28
Degree $2$
Conductor $363$
Sign $0.626 + 0.779i$
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  + (1.79 − 2.47i)2-s + (−2.81 − 1.04i)3-s + (−1.65 − 5.10i)4-s + (3.90 + 5.37i)5-s + (−7.64 + 5.07i)6-s + (0.946 + 2.91i)7-s + (−3.97 − 1.29i)8-s + (6.80 + 5.88i)9-s + 20.3·10-s + (−0.678 + 16.0i)12-s + (13.1 + 9.54i)13-s + (8.91 + 2.89i)14-s + (−5.35 − 19.1i)15-s + (7.01 − 5.09i)16-s + (0.473 + 0.651i)17-s + (26.8 − 6.27i)18-s + ⋯
L(s)  = 1  + (0.899 − 1.23i)2-s + (−0.937 − 0.348i)3-s + (−0.414 − 1.27i)4-s + (0.780 + 1.07i)5-s + (−1.27 + 0.846i)6-s + (0.135 + 0.416i)7-s + (−0.496 − 0.161i)8-s + (0.756 + 0.653i)9-s + 2.03·10-s + (−0.0565 + 1.34i)12-s + (1.01 + 0.734i)13-s + (0.636 + 0.206i)14-s + (−0.356 − 1.27i)15-s + (0.438 − 0.318i)16-s + (0.0278 + 0.0383i)17-s + (1.48 − 0.348i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(363\)    =    \(3 \cdot 11^{2}\)
Sign: $0.626 + 0.779i$
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: \((2,\ 363,\ (\ :1),\ 0.626 + 0.779i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.19123 - 1.04920i\)
\(L(\frac12)\) \(\approx\) \(2.19123 - 1.04920i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.81 + 1.04i)T \)
11 \( 1 \)
good2 \( 1 + (-1.79 + 2.47i)T + (-1.23 - 3.80i)T^{2} \)
5 \( 1 + (-3.90 - 5.37i)T + (-7.72 + 23.7i)T^{2} \)
7 \( 1 + (-0.946 - 2.91i)T + (-39.6 + 28.8i)T^{2} \)
13 \( 1 + (-13.1 - 9.54i)T + (52.2 + 160. i)T^{2} \)
17 \( 1 + (-0.473 - 0.651i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (6.39 - 19.6i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + 27.3iT - 529T^{2} \)
29 \( 1 + (-3.59 + 1.16i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (-16.8 - 12.2i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (-11.9 - 36.6i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (12.7 + 4.13i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 + 43.4T + 1.84e3T^{2} \)
47 \( 1 + (18.9 + 6.14i)T + (1.78e3 + 1.29e3i)T^{2} \)
53 \( 1 + (-10.3 + 14.2i)T + (-868. - 2.67e3i)T^{2} \)
59 \( 1 + (-41.3 + 13.4i)T + (2.81e3 - 2.04e3i)T^{2} \)
61 \( 1 + (-8.61 + 6.26i)T + (1.14e3 - 3.53e3i)T^{2} \)
67 \( 1 - 72.2T + 4.48e3T^{2} \)
71 \( 1 + (-1.51 - 2.07i)T + (-1.55e3 + 4.79e3i)T^{2} \)
73 \( 1 + (-14.0 - 43.2i)T + (-4.31e3 + 3.13e3i)T^{2} \)
79 \( 1 + (79.4 + 57.7i)T + (1.92e3 + 5.93e3i)T^{2} \)
83 \( 1 + (-18.7 - 25.8i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + 18.5iT - 7.92e3T^{2} \)
97 \( 1 + (-51.2 - 37.2i)T + (2.90e3 + 8.94e3i)T^{2} \)
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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

−11.27418476300218776306910634222, −10.41484694495668090538812739937, −10.00486705110273345543315943850, −8.334346261318151754440170794091, −6.71722111862354305280502355567, −6.12716798278060476876154666570, −5.08050984488687304172680877428, −3.87785105726394902401524654355, −2.50303742478640828903222311717, −1.52558812653928729788333337509, 1.10118123998169449971017074445, 3.81624782654864627654612129835, 4.81929110885775511328374865985, 5.49538638630367734672606623595, 6.18241178060955993092495032064, 7.19811227065998058169211744930, 8.365119531079817714688309644960, 9.423108053971373720828977831665, 10.46231859380691702995876062943, 11.44715117828321701604920157292

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