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Properties

Label 2-369-9.4-c1-0-36
Degree $2$
Conductor $369$
Sign $-0.986 - 0.163i$
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  + (−0.373 − 0.646i)2-s + (−0.160 − 1.72i)3-s + (0.721 − 1.24i)4-s + (−1.18 + 2.04i)5-s + (−1.05 + 0.747i)6-s + (0.193 + 0.334i)7-s − 2.56·8-s + (−2.94 + 0.552i)9-s + 1.76·10-s + (−2.61 − 4.52i)11-s + (−2.27 − 1.04i)12-s + (0.822 − 1.42i)13-s + (0.144 − 0.249i)14-s + (3.71 + 1.70i)15-s + (−0.484 − 0.838i)16-s − 6.42·17-s + ⋯
L(s)  = 1  + (−0.263 − 0.456i)2-s + (−0.0925 − 0.995i)3-s + (0.360 − 0.624i)4-s + (−0.528 + 0.915i)5-s + (−0.430 + 0.304i)6-s + (0.0729 + 0.126i)7-s − 0.908·8-s + (−0.982 + 0.184i)9-s + 0.557·10-s + (−0.787 − 1.36i)11-s + (−0.655 − 0.301i)12-s + (0.228 − 0.395i)13-s + (0.0385 − 0.0667i)14-s + (0.960 + 0.441i)15-s + (−0.121 − 0.209i)16-s − 1.55·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.163i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $-0.986 - 0.163i$
Analytic conductor: \(2.94647\)
Root analytic conductor: \(1.71653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1/2),\ -0.986 - 0.163i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0540838 + 0.659058i\)
\(L(\frac12)\) \(\approx\) \(0.0540838 + 0.659058i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.160 + 1.72i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.373 + 0.646i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (1.18 - 2.04i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.193 - 0.334i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.61 + 4.52i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.822 + 1.42i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.42T + 17T^{2} \)
19 \( 1 + 3.07T + 19T^{2} \)
23 \( 1 + (-3.88 + 6.73i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.70 - 8.14i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.19 + 2.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4.99T + 37T^{2} \)
43 \( 1 + (2.80 + 4.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.83 + 4.90i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 1.07T + 53T^{2} \)
59 \( 1 + (-3.07 + 5.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.78 + 8.28i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.78 + 4.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.9T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + (-3.37 - 5.84i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.388 + 0.673i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 7.01T + 89T^{2} \)
97 \( 1 + (-3.91 - 6.78i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.84747754002285675098546078649, −10.63864560701946791830408963404, −8.833672864662416876102416815141, −8.257427700417107211250279123761, −6.82990806954420064323069756481, −6.44293686925277708279316941385, −5.23699045660914557840235603401, −3.14330802278140980634825800207, −2.33909019527947211290884007322, −0.45783767292706726772607802352, 2.59184991744776144602601886357, 4.21984733787412630959778116484, 4.70243260144409285976424531455, 6.18209899875759285073434141654, 7.34908515355649000998761771378, 8.289229442211387680484301167573, 8.981410407430992548294664707938, 9.862524110258723558263663920400, 11.07443957396681617745128896013, 11.74126152968078163952954239492

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