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Properties

Label 2-369-41.38-c2-0-33
Degree $2$
Conductor $369$
Sign $-0.855 - 0.517i$
Analytic cond. $10.0545$
Root an. cond. $3.17088$
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  + (2.38 − 2.38i)2-s − 7.36i·4-s + (−2.78 − 2.78i)5-s + (−9.79 − 4.05i)7-s + (−8.02 − 8.02i)8-s − 13.2·10-s + (−4.61 + 11.1i)11-s + (−2.11 − 0.877i)13-s + (−33.0 + 13.6i)14-s − 8.79·16-s + (2.83 − 1.17i)17-s + (8.73 − 3.61i)19-s + (−20.4 + 20.4i)20-s + (15.5 + 37.5i)22-s − 20.5i·23-s + ⋯
L(s)  = 1  + (1.19 − 1.19i)2-s − 1.84i·4-s + (−0.556 − 0.556i)5-s + (−1.39 − 0.579i)7-s + (−1.00 − 1.00i)8-s − 1.32·10-s + (−0.419 + 1.01i)11-s + (−0.162 − 0.0674i)13-s + (−2.35 + 0.976i)14-s − 0.549·16-s + (0.166 − 0.0690i)17-s + (0.459 − 0.190i)19-s + (−1.02 + 1.02i)20-s + (0.706 + 1.70i)22-s − 0.892i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $-0.855 - 0.517i$
Analytic conductor: \(10.0545\)
Root analytic conductor: \(3.17088\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1),\ -0.855 - 0.517i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.446368 + 1.59921i\)
\(L(\frac12)\) \(\approx\) \(0.446368 + 1.59921i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 + (-6.99 + 40.3i)T \)
good2 \( 1 + (-2.38 + 2.38i)T - 4iT^{2} \)
5 \( 1 + (2.78 + 2.78i)T + 25iT^{2} \)
7 \( 1 + (9.79 + 4.05i)T + (34.6 + 34.6i)T^{2} \)
11 \( 1 + (4.61 - 11.1i)T + (-85.5 - 85.5i)T^{2} \)
13 \( 1 + (2.11 + 0.877i)T + (119. + 119. i)T^{2} \)
17 \( 1 + (-2.83 + 1.17i)T + (204. - 204. i)T^{2} \)
19 \( 1 + (-8.73 + 3.61i)T + (255. - 255. i)T^{2} \)
23 \( 1 + 20.5iT - 529T^{2} \)
29 \( 1 + (8.67 + 3.59i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 33.7iT - 961T^{2} \)
37 \( 1 + 60.1T + 1.36e3T^{2} \)
43 \( 1 + (-6.83 + 6.83i)T - 1.84e3iT^{2} \)
47 \( 1 + (0.199 - 0.0824i)T + (1.56e3 - 1.56e3i)T^{2} \)
53 \( 1 + (-6.93 + 16.7i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 - 36.2T + 3.48e3T^{2} \)
61 \( 1 + (81.0 - 81.0i)T - 3.72e3iT^{2} \)
67 \( 1 + (-49.6 + 20.5i)T + (3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-13.8 - 5.72i)T + (3.56e3 + 3.56e3i)T^{2} \)
73 \( 1 + (-47.3 + 47.3i)T - 5.32e3iT^{2} \)
79 \( 1 + (-14.7 + 35.6i)T + (-4.41e3 - 4.41e3i)T^{2} \)
83 \( 1 - 99.1T + 6.88e3T^{2} \)
89 \( 1 + (-40.3 - 16.7i)T + (5.60e3 + 5.60e3i)T^{2} \)
97 \( 1 + (-50.8 - 122. i)T + (-6.65e3 + 6.65e3i)T^{2} \)
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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

−10.67696400626929205826974931677, −10.11523431303862510104395315683, −9.225079575526069470021728874456, −7.68559684845339098336268375886, −6.58142262550748763894499627377, −5.30469893534831877056922183527, −4.32955724217386358636279528129, −3.52987266300196875187283097358, −2.33127413820066340056016209090, −0.48578636895767129725915614114, 3.18428409395025550000467400576, 3.50366937994363278393499881697, 5.16302708518680140072883579058, 5.95529476906472501094182530357, 6.78938001030234420605569217319, 7.57939422432800589437853884624, 8.632113795149441451524586768785, 9.792629133182408977538673961526, 11.04525114063572835345793780013, 12.10094563876904561310415824464

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