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Properties

Label 2-494-13.3-c1-0-11
Degree $2$
Conductor $494$
Sign $-0.500 + 0.866i$
Analytic cond. $3.94460$
Root an. cond. $1.98610$
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 + (−0.825 − 1.42i)3-s + (−0.499 + 0.866i)4-s + 1.27·5-s + (−0.825 + 1.42i)6-s + (1.32 − 2.29i)7-s + 0.999·8-s + (0.136 − 0.237i)9-s + (−0.636 − 1.10i)10-s + (1.63 + 2.83i)11-s + 1.65·12-s + (3.47 + 0.955i)13-s − 2.65·14-s + (−1.05 − 1.82i)15-s + (−0.5 − 0.866i)16-s + (2.65 − 4.59i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.476 − 0.825i)3-s + (−0.249 + 0.433i)4-s + 0.569·5-s + (−0.337 + 0.583i)6-s + (0.501 − 0.867i)7-s + 0.353·8-s + (0.0456 − 0.0790i)9-s + (−0.201 − 0.348i)10-s + (0.493 + 0.854i)11-s + 0.476·12-s + (0.964 + 0.265i)13-s − 0.708·14-s + (−0.271 − 0.470i)15-s + (−0.125 − 0.216i)16-s + (0.642 − 1.11i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.500 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(494\)    =    \(2 \cdot 13 \cdot 19\)
Sign: $-0.500 + 0.866i$
Analytic conductor: \(3.94460\)
Root analytic conductor: \(1.98610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{494} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 494,\ (\ :1/2),\ -0.500 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.600814 - 1.04064i\)
\(L(\frac12)\) \(\approx\) \(0.600814 - 1.04064i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-3.47 - 0.955i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (0.825 + 1.42i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 1.27T + 5T^{2} \)
7 \( 1 + (-1.32 + 2.29i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.63 - 2.83i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-2.65 + 4.59i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (3.20 + 5.54i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.74 - 6.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.47T + 31T^{2} \)
37 \( 1 + (1.10 + 1.91i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.94 + 3.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.15 + 5.45i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.3T + 47T^{2} \)
53 \( 1 + 2.47T + 53T^{2} \)
59 \( 1 + (4.76 - 8.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.83 + 4.91i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.103 - 0.178i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 9.41T + 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 - 7.79T + 83T^{2} \)
89 \( 1 + (-7.06 - 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.65 + 9.78i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.63438307109752776026754979945, −9.932676115416885990990582786595, −9.002719549930740613395485996087, −7.85850417413814294668294518624, −7.01198228357130810195142577652, −6.23664052015485833839751196015, −4.82907222271004048403301472819, −3.70595863165484345813031614096, −1.95738407178373122195214168403, −0.989702654515668532415860043930, 1.66787909024400568107032440461, 3.62240325295579208205728752692, 4.85308438870163635451462136334, 5.97902843937601850588384166492, 6.05695604471239873389323342607, 7.955927138363930377184934391487, 8.464550918522004966509673594861, 9.590275019869082950542633266305, 10.13779954655201656803254785539, 11.17487885630373446510194194972

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