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Properties

Label 2-1254-209.164-c1-0-38
Degree $2$
Conductor $1254$
Sign $-0.662 + 0.749i$
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 + (−2.02 − 3.51i)5-s + (0.866 + 0.499i)6-s − 4.70i·7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (2.02 − 3.51i)10-s + (3.26 + 0.586i)11-s + 0.999i·12-s + (−2.31 + 4.00i)13-s + (4.07 − 2.35i)14-s + (−3.51 − 2.02i)15-s + (−0.5 − 0.866i)16-s + (−5.05 + 2.91i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 + 0.433i)4-s + (−0.907 − 1.57i)5-s + (0.353 + 0.204i)6-s − 1.77i·7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.641 − 1.11i)10-s + (0.984 + 0.176i)11-s + 0.288i·12-s + (−0.641 + 1.11i)13-s + (1.08 − 0.628i)14-s + (−0.907 − 0.524i)15-s + (−0.125 − 0.216i)16-s + (−1.22 + 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1254\)    =    \(2 \cdot 3 \cdot 11 \cdot 19\)
Sign: $-0.662 + 0.749i$
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: \((2,\ 1254,\ (\ :1/2),\ -0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.202291938\)
\(L(\frac12)\) \(\approx\) \(1.202291938\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.26 - 0.586i)T \)
19 \( 1 + (2.45 + 3.60i)T \)
good5 \( 1 + (2.02 + 3.51i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 4.70iT - 7T^{2} \)
13 \( 1 + (2.31 - 4.00i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.05 - 2.91i)T + (8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.59 - 2.76i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.36 + 2.36i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.59iT - 31T^{2} \)
37 \( 1 - 1.45iT - 37T^{2} \)
41 \( 1 + (-4.75 - 8.23i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.64 - 2.10i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.09 + 8.82i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.14 + 2.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.8 + 6.28i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (3.54 + 2.04i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.89 + 3.40i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (8.80 - 5.08i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.57 - 0.911i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.102 - 0.178i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.68iT - 83T^{2} \)
89 \( 1 + (-4.02 - 2.32i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.58 + 4.37i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.151725229006495123365397254549, −8.463832038682161480271697382215, −7.69783388688278032203890577371, −7.05822036602162502000685555728, −6.32548011312406240252826621803, −4.60279868666591207217774979123, −4.35285494493473140131580780177, −3.78658521585198734642407581913, −1.74139754749872508797152729540, −0.41053439907209740566583223815, 2.28148750034893453448267701432, 2.82277621391978948552394795968, 3.61661027861633467378058952027, 4.65134100354700970542143126190, 5.82154217204122815878618621400, 6.61010992086905761391266725266, 7.56888102564950155701136544008, 8.635018686486374899669824983256, 9.073870347319548082720430535252, 10.26082837180118375566189100149

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