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Properties

Label 2-494-13.3-c1-0-9
Degree $2$
Conductor $494$
Sign $1$
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 + (0.636 + 1.10i)3-s + (−0.499 + 0.866i)4-s + 2.37·5-s + (0.636 − 1.10i)6-s + (−0.136 + 0.237i)7-s + 0.999·8-s + (0.688 − 1.19i)9-s + (−1.18 − 2.05i)10-s + (2.18 + 3.79i)11-s − 1.27·12-s + (−0.910 − 3.48i)13-s + 0.273·14-s + (1.51 + 2.62i)15-s + (−0.5 − 0.866i)16-s + (−0.273 + 0.474i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.367 + 0.636i)3-s + (−0.249 + 0.433i)4-s + 1.06·5-s + (0.260 − 0.450i)6-s + (−0.0517 + 0.0896i)7-s + 0.353·8-s + (0.229 − 0.397i)9-s + (−0.375 − 0.651i)10-s + (0.659 + 1.14i)11-s − 0.367·12-s + (−0.252 − 0.967i)13-s + 0.0732·14-s + (0.390 + 0.677i)15-s + (−0.125 − 0.216i)16-s + (−0.0664 + 0.115i)17-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(494\)    =    \(2 \cdot 13 \cdot 19\)
Sign: $1$
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: \((2,\ 494,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62242\)
\(L(\frac12)\) \(\approx\) \(1.62242\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.910 + 3.48i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.636 - 1.10i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.37T + 5T^{2} \)
7 \( 1 + (0.136 - 0.237i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.18 - 3.79i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.273 - 0.474i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-2.28 - 3.96i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.839 + 1.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.85T + 31T^{2} \)
37 \( 1 + (-4.02 - 6.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.51 + 7.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.226 + 0.391i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 0.472T + 47T^{2} \)
53 \( 1 + 2.85T + 53T^{2} \)
59 \( 1 + (-6.76 + 11.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.09 - 3.63i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.02 + 8.70i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.1T + 73T^{2} \)
79 \( 1 - 2.36T + 79T^{2} \)
83 \( 1 + 16.7T + 83T^{2} \)
89 \( 1 + (-1.02 - 1.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.72 + 4.72i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.59474024275136987438486731830, −9.825502266854604758646765116615, −9.579357216020618214307860115577, −8.640140765737816419644374383118, −7.42914240717104921954666835654, −6.31419598233003830530767937158, −5.08781843006310068438998610125, −3.99838625918314549078234000235, −2.83657690662771356618343110596, −1.54788216077619190983167440981, 1.34067665493308246955531617136, 2.57246235185003862638730411376, 4.36615591258367302464679870779, 5.62425647437289777652342745597, 6.51281513085540320981353397578, 7.15896824044308883883910194183, 8.353826818488406667266220823781, 8.992131423010659723770915296640, 9.848469474235158581823613640611, 10.78145480244895508150962513258

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