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Properties

Label 2-369-41.5-c1-0-9
Degree $2$
Conductor $369$
Sign $0.466 - 0.884i$
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  + (1.00 + 1.38i)2-s + (−0.290 + 0.892i)4-s + (2.57 + 0.837i)5-s + (−0.252 − 1.59i)7-s + (1.73 − 0.562i)8-s + (1.43 + 4.41i)10-s + (−0.314 + 0.617i)11-s + (−1.50 − 0.238i)13-s + (1.95 − 1.95i)14-s + (4.04 + 2.93i)16-s + (−0.942 − 0.480i)17-s + (−0.126 + 0.0201i)19-s + (−1.49 + 2.05i)20-s + (−1.17 + 0.185i)22-s + (−1.52 + 1.10i)23-s + ⋯
L(s)  = 1  + (0.712 + 0.980i)2-s + (−0.145 + 0.446i)4-s + (1.15 + 0.374i)5-s + (−0.0953 − 0.602i)7-s + (0.611 − 0.198i)8-s + (0.453 + 1.39i)10-s + (−0.0948 + 0.186i)11-s + (−0.418 − 0.0662i)13-s + (0.522 − 0.522i)14-s + (1.01 + 0.734i)16-s + (−0.228 − 0.116i)17-s + (−0.0291 + 0.00461i)19-s + (−0.334 + 0.460i)20-s + (−0.250 + 0.0396i)22-s + (−0.317 + 0.230i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.466 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $0.466 - 0.884i$
Analytic conductor: \(2.94647\)
Root analytic conductor: \(1.71653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (46, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1/2),\ 0.466 - 0.884i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97765 + 1.19256i\)
\(L(\frac12)\) \(\approx\) \(1.97765 + 1.19256i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.96 - 2.31i)T \)
good2 \( 1 + (-1.00 - 1.38i)T + (-0.618 + 1.90i)T^{2} \)
5 \( 1 + (-2.57 - 0.837i)T + (4.04 + 2.93i)T^{2} \)
7 \( 1 + (0.252 + 1.59i)T + (-6.65 + 2.16i)T^{2} \)
11 \( 1 + (0.314 - 0.617i)T + (-6.46 - 8.89i)T^{2} \)
13 \( 1 + (1.50 + 0.238i)T + (12.3 + 4.01i)T^{2} \)
17 \( 1 + (0.942 + 0.480i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (0.126 - 0.0201i)T + (18.0 - 5.87i)T^{2} \)
23 \( 1 + (1.52 - 1.10i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (7.39 - 3.76i)T + (17.0 - 23.4i)T^{2} \)
31 \( 1 + (-0.373 - 1.14i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.86 - 5.75i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (6.21 + 8.55i)T + (-13.2 + 40.8i)T^{2} \)
47 \( 1 + (-0.991 + 6.26i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-11.1 + 5.67i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-6.61 + 4.80i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.54 - 2.13i)T + (-18.8 - 58.0i)T^{2} \)
67 \( 1 + (3.85 + 7.56i)T + (-39.3 + 54.2i)T^{2} \)
71 \( 1 + (1.94 - 3.82i)T + (-41.7 - 57.4i)T^{2} \)
73 \( 1 - 12.1iT - 73T^{2} \)
79 \( 1 + (8.14 + 8.14i)T + 79iT^{2} \)
83 \( 1 + 6.49T + 83T^{2} \)
89 \( 1 + (0.451 + 2.84i)T + (-84.6 + 27.5i)T^{2} \)
97 \( 1 + (-0.548 - 1.07i)T + (-57.0 + 78.4i)T^{2} \)
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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

−11.61042050173966022022324391622, −10.24965719955952781864995759277, −10.07237906308384138714661216580, −8.644209621977401595635222871682, −7.28932994884307715580325262982, −6.78966252811185749965703956765, −5.73677971768911913394827530716, −5.00404073022108369008872458420, −3.67128790212545094492407724265, −1.91714425097103228458875477499, 1.79784077715225519159350378738, 2.68695474205412405481331826936, 4.10139118935323378969529113160, 5.27362922727123913480690364860, 5.99361210398857342901045152101, 7.46946616394380806351188527197, 8.738968734026138536784903906869, 9.646194362549715158089392784361, 10.42252464852175198351938897351, 11.43259217536843506487130678879

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