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Properties

Label 2-1254-209.164-c1-0-4
Degree $2$
Conductor $1254$
Sign $-0.989 + 0.144i$
Analytic cond. $10.0132$
Root an. cond. $3.16437$
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  + (0.5 + 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (1.37 + 2.38i)5-s + (−0.866 − 0.499i)6-s − 4.25i·7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (−1.37 + 2.38i)10-s + (1.41 − 2.99i)11-s − 0.999i·12-s + (−3.47 + 6.01i)13-s + (3.68 − 2.12i)14-s + (−2.38 − 1.37i)15-s + (−0.5 − 0.866i)16-s + (−4.76 + 2.75i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.616 + 1.06i)5-s + (−0.353 − 0.204i)6-s − 1.60i·7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (−0.436 + 0.755i)10-s + (0.428 − 0.903i)11-s − 0.288i·12-s + (−0.963 + 1.66i)13-s + (0.984 − 0.568i)14-s + (−0.616 − 0.356i)15-s + (−0.125 − 0.216i)16-s + (−1.15 + 0.667i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.989 + 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1254\)    =    \(2 \cdot 3 \cdot 11 \cdot 19\)
Sign: $-0.989 + 0.144i$
Analytic conductor: \(10.0132\)
Root analytic conductor: \(3.16437\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1254} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1254,\ (\ :1/2),\ -0.989 + 0.144i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9129601683\)
\(L(\frac12)\) \(\approx\) \(0.9129601683\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
11 \( 1 + (-1.41 + 2.99i)T \)
19 \( 1 + (4.30 - 0.701i)T \)
good5 \( 1 + (-1.37 - 2.38i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 4.25iT - 7T^{2} \)
13 \( 1 + (3.47 - 6.01i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.76 - 2.75i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.882 - 1.52i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.07 - 7.05i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.97iT - 31T^{2} \)
37 \( 1 - 7.57iT - 37T^{2} \)
41 \( 1 + (4.84 + 8.39i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.39 - 1.38i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.20 + 3.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.984 - 0.568i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.65 - 1.53i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-11.1 - 6.42i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.9 + 6.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.40 + 2.54i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-11.1 + 6.44i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.97 - 5.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.905iT - 83T^{2} \)
89 \( 1 + (-1.05 - 0.611i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.40 - 0.812i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.41717827535281218882915997708, −9.310976285287791585999536106386, −8.505663343111697291214898151851, −7.04187704850927400612589989430, −6.85767728187999062728156192960, −6.28721758043976741192572348187, −5.00291537834277050857006363413, −4.14925301142443574858900609499, −3.42996713135243143183525881842, −1.82486052103613800025461970110, 0.33679317998574431107465008220, 2.09366429212032356962769614245, 2.45174613173152441414172513208, 4.32696395589164028676325436364, 5.09416267746237920998709848900, 5.65311721548248203736127285276, 6.42023877168418437257718035828, 7.76356280462216042656148619501, 8.666285721478665415571742056069, 9.491482357568476707427523526569

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