Properties

Label 2-369-41.20-c1-0-5
Degree $2$
Conductor $369$
Sign $0.477 + 0.878i$
Analytic cond. $2.94647$
Root an. cond. $1.71653$
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  + (−1.14 − 0.372i)2-s + (−0.439 − 0.319i)4-s + (−1.49 + 2.05i)5-s + (−1.01 − 0.517i)7-s + (1.80 + 2.48i)8-s + (2.48 − 1.80i)10-s + (0.807 − 5.10i)11-s + (1.17 + 2.30i)13-s + (0.972 + 0.972i)14-s + (−0.809 − 2.48i)16-s + (3.05 + 0.483i)17-s + (3.74 − 7.34i)19-s + (1.31 − 0.427i)20-s + (−2.82 + 5.55i)22-s + (1.17 − 3.62i)23-s + ⋯
L(s)  = 1  + (−0.811 − 0.263i)2-s + (−0.219 − 0.159i)4-s + (−0.669 + 0.920i)5-s + (−0.383 − 0.195i)7-s + (0.637 + 0.878i)8-s + (0.785 − 0.570i)10-s + (0.243 − 1.53i)11-s + (0.325 + 0.638i)13-s + (0.259 + 0.259i)14-s + (−0.202 − 0.622i)16-s + (0.741 + 0.117i)17-s + (0.858 − 1.68i)19-s + (0.294 − 0.0955i)20-s + (−0.603 + 1.18i)22-s + (0.245 − 0.756i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $0.477 + 0.878i$
Analytic conductor: \(2.94647\)
Root analytic conductor: \(1.71653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1/2),\ 0.477 + 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.554209 - 0.329386i\)
\(L(\frac12)\) \(\approx\) \(0.554209 - 0.329386i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 + (-5.78 + 2.74i)T \)
good2 \( 1 + (1.14 + 0.372i)T + (1.61 + 1.17i)T^{2} \)
5 \( 1 + (1.49 - 2.05i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.01 + 0.517i)T + (4.11 + 5.66i)T^{2} \)
11 \( 1 + (-0.807 + 5.10i)T + (-10.4 - 3.39i)T^{2} \)
13 \( 1 + (-1.17 - 2.30i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-3.05 - 0.483i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (-3.74 + 7.34i)T + (-11.1 - 15.3i)T^{2} \)
23 \( 1 + (-1.17 + 3.62i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (5.72 - 0.906i)T + (27.5 - 8.96i)T^{2} \)
31 \( 1 + (-3.65 + 2.65i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.584 - 0.424i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-1.75 - 0.570i)T + (34.7 + 25.2i)T^{2} \)
47 \( 1 + (-3.15 + 1.60i)T + (27.6 - 38.0i)T^{2} \)
53 \( 1 + (-5.52 + 0.875i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (0.242 - 0.747i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-6.70 + 2.17i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + (0.835 + 5.27i)T + (-63.7 + 20.7i)T^{2} \)
71 \( 1 + (0.708 - 4.47i)T + (-67.5 - 21.9i)T^{2} \)
73 \( 1 + 0.149iT - 73T^{2} \)
79 \( 1 + (11.4 - 11.4i)T - 79iT^{2} \)
83 \( 1 + 4.27T + 83T^{2} \)
89 \( 1 + (-1.51 - 0.773i)T + (52.3 + 72.0i)T^{2} \)
97 \( 1 + (-0.687 - 4.34i)T + (-92.2 + 29.9i)T^{2} \)
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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

−11.27380845405936189148665308478, −10.39562846615734463474092319845, −9.350112097778056173702221056235, −8.650311999015554548186064063506, −7.61431496116755494852832768171, −6.68415934062411875737151509204, −5.46247769716910793933452120607, −3.95061093273281437969891046867, −2.83066638906315903968872515779, −0.71130095507330376817275724638, 1.23486999940708928116135741858, 3.56027771797540171871892626561, 4.53344943104453299893458326019, 5.76061103962340354432540234120, 7.38194942749866049681587225861, 7.78421760343106549397700603973, 8.774996171614572927227121058935, 9.663300032960063197252138541496, 10.16379564962613239083934029691, 11.73954577420670843772766108947

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