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Properties

Label 2-369-41.8-c1-0-15
Degree $2$
Conductor $369$
Sign $-0.872 + 0.489i$
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  + (1.61 − 2.21i)2-s + (−1.70 − 5.23i)4-s + (1.15 − 0.375i)5-s + (1.19 + 0.189i)7-s + (−9.13 − 2.96i)8-s + (1.02 − 3.16i)10-s + (−0.836 + 0.426i)11-s + (0.289 + 1.82i)13-s + (2.34 − 2.34i)14-s + (−12.3 + 8.98i)16-s + (1.02 + 2.01i)17-s + (0.815 − 5.15i)19-s + (−3.92 − 5.40i)20-s + (−0.402 + 2.53i)22-s + (5.31 + 3.86i)23-s + ⋯
L(s)  = 1  + (1.13 − 1.56i)2-s + (−0.850 − 2.61i)4-s + (0.516 − 0.167i)5-s + (0.451 + 0.0715i)7-s + (−3.22 − 1.04i)8-s + (0.324 − 0.999i)10-s + (−0.252 + 0.128i)11-s + (0.0802 + 0.506i)13-s + (0.626 − 0.626i)14-s + (−3.09 + 2.24i)16-s + (0.248 + 0.488i)17-s + (0.187 − 1.18i)19-s + (−0.878 − 1.20i)20-s + (−0.0857 + 0.541i)22-s + (1.10 + 0.804i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $-0.872 + 0.489i$
Analytic conductor: \(2.94647\)
Root analytic conductor: \(1.71653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (172, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1/2),\ -0.872 + 0.489i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.606244 - 2.31943i\)
\(L(\frac12)\) \(\approx\) \(0.606244 - 2.31943i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.89 + 4.13i)T \)
good2 \( 1 + (-1.61 + 2.21i)T + (-0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.15 + 0.375i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.19 - 0.189i)T + (6.65 + 2.16i)T^{2} \)
11 \( 1 + (0.836 - 0.426i)T + (6.46 - 8.89i)T^{2} \)
13 \( 1 + (-0.289 - 1.82i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-1.02 - 2.01i)T + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.815 + 5.15i)T + (-18.0 - 5.87i)T^{2} \)
23 \( 1 + (-5.31 - 3.86i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.689 + 1.35i)T + (-17.0 - 23.4i)T^{2} \)
31 \( 1 + (0.515 - 1.58i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.97 - 6.06i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-0.714 + 0.982i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (-5.36 + 0.849i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (-4.56 + 8.95i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (-6.23 - 4.52i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.982 - 1.35i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (-13.1 - 6.68i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (2.77 - 1.41i)T + (41.7 - 57.4i)T^{2} \)
73 \( 1 + 0.596iT - 73T^{2} \)
79 \( 1 + (5.89 + 5.89i)T + 79iT^{2} \)
83 \( 1 + 11.1T + 83T^{2} \)
89 \( 1 + (6.08 + 0.964i)T + (84.6 + 27.5i)T^{2} \)
97 \( 1 + (15.0 + 7.65i)T + (57.0 + 78.4i)T^{2} \)
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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

−11.35695096565369141489450807675, −10.33859214522982078896561517263, −9.585147067518682935991599036020, −8.701780970534523064660517418844, −6.86512333399711949022104143661, −5.56006256942505168290464962390, −4.94676382284665694064939734260, −3.78442389981262926047043986043, −2.54215294054860803950824517682, −1.38674677402854595389416115088, 2.84270870674236314791138639508, 4.10653869313498676236055908251, 5.22303900403702319819285742791, 5.88076314690616194018756623118, 6.88942017519357920897065172825, 7.81422142501176376716906283448, 8.514949566150698608363584103881, 9.730393710539046819840373884118, 11.08412040022952720210247277191, 12.25753094435058187815835820750

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