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Properties

Label 2-369-41.8-c1-0-10
Degree $2$
Conductor $369$
Sign $0.658 - 0.752i$
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  + (−0.746 + 1.02i)2-s + (0.119 + 0.366i)4-s + (3.27 − 1.06i)5-s + (1.70 + 0.270i)7-s + (−2.88 − 0.936i)8-s + (−1.35 + 4.16i)10-s + (3.99 − 2.03i)11-s + (−0.380 − 2.40i)13-s + (−1.55 + 1.55i)14-s + (2.49 − 1.81i)16-s + (0.138 + 0.272i)17-s + (0.274 − 1.73i)19-s + (0.781 + 1.07i)20-s + (−0.890 + 5.62i)22-s + (−3.75 − 2.72i)23-s + ⋯
L(s)  = 1  + (−0.528 + 0.726i)2-s + (0.0596 + 0.183i)4-s + (1.46 − 0.476i)5-s + (0.644 + 0.102i)7-s + (−1.01 − 0.331i)8-s + (−0.428 + 1.31i)10-s + (1.20 − 0.613i)11-s + (−0.105 − 0.665i)13-s + (−0.414 + 0.414i)14-s + (0.622 − 0.452i)16-s + (0.0336 + 0.0660i)17-s + (0.0629 − 0.397i)19-s + (0.174 + 0.240i)20-s + (−0.189 + 1.19i)22-s + (−0.782 − 0.568i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $0.658 - 0.752i$
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: \((2,\ 369,\ (\ :1/2),\ 0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29508 + 0.587361i\)
\(L(\frac12)\) \(\approx\) \(1.29508 + 0.587361i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.99 - 2.26i)T \)
good2 \( 1 + (0.746 - 1.02i)T + (-0.618 - 1.90i)T^{2} \)
5 \( 1 + (-3.27 + 1.06i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (-1.70 - 0.270i)T + (6.65 + 2.16i)T^{2} \)
11 \( 1 + (-3.99 + 2.03i)T + (6.46 - 8.89i)T^{2} \)
13 \( 1 + (0.380 + 2.40i)T + (-12.3 + 4.01i)T^{2} \)
17 \( 1 + (-0.138 - 0.272i)T + (-9.99 + 13.7i)T^{2} \)
19 \( 1 + (-0.274 + 1.73i)T + (-18.0 - 5.87i)T^{2} \)
23 \( 1 + (3.75 + 2.72i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (4.39 - 8.62i)T + (-17.0 - 23.4i)T^{2} \)
31 \( 1 + (2.28 - 7.03i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-2.28 - 7.02i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (1.53 - 2.10i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (6.25 - 0.991i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (-0.556 + 1.09i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (2.66 + 1.93i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.655 + 0.901i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (9.09 + 4.63i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (0.475 - 0.242i)T + (41.7 - 57.4i)T^{2} \)
73 \( 1 + 8.56iT - 73T^{2} \)
79 \( 1 + (6.09 + 6.09i)T + 79iT^{2} \)
83 \( 1 + 9.88T + 83T^{2} \)
89 \( 1 + (-14.4 - 2.28i)T + (84.6 + 27.5i)T^{2} \)
97 \( 1 + (-8.80 - 4.48i)T + (57.0 + 78.4i)T^{2} \)
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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

−11.52786374858274152647225182169, −10.39130073600496626871523756545, −9.302776652021480445931852102488, −8.836285888368543397360834453793, −7.923465917081476757592630412396, −6.65975724416378263883064766388, −5.97946763858326866029167928609, −4.94178841380373779848299724960, −3.19045809158314058203441736359, −1.51581170950761631585000010167, 1.63648645157766239969615566797, 2.25814052960957589432801860237, 4.09019905770861707744800241135, 5.69673762475622332629485370035, 6.27959982158412109312978528644, 7.53544031849123277289890761510, 9.050556165832444003531037475933, 9.621872666326731964735823531099, 10.15147225907529335301012685233, 11.30548698363928894607618621680

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