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Properties

Label 2-494-13.10-c1-0-13
Degree $2$
Conductor $494$
Sign $0.999 - 0.0329i$
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  + (0.866 − 0.5i)2-s + (1.32 + 2.29i)3-s + (0.499 − 0.866i)4-s − 3.69i·5-s + (2.29 + 1.32i)6-s + (3.52 + 2.03i)7-s − 0.999i·8-s + (−1.99 + 3.46i)9-s + (−1.84 − 3.19i)10-s + (−4.94 + 2.85i)11-s + 2.64·12-s + (3.03 − 1.94i)13-s + 4.07·14-s + (8.45 − 4.88i)15-s + (−0.5 − 0.866i)16-s + (1.68 − 2.91i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.763 + 1.32i)3-s + (0.249 − 0.433i)4-s − 1.65i·5-s + (0.935 + 0.539i)6-s + (1.33 + 0.769i)7-s − 0.353i·8-s + (−0.666 + 1.15i)9-s + (−0.583 − 1.01i)10-s + (−1.49 + 0.860i)11-s + 0.763·12-s + (0.842 − 0.539i)13-s + 1.08·14-s + (2.18 − 1.26i)15-s + (−0.125 − 0.216i)16-s + (0.408 − 0.706i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(494\)    =    \(2 \cdot 13 \cdot 19\)
Sign: $0.999 - 0.0329i$
Analytic conductor: \(3.94460\)
Root analytic conductor: \(1.98610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{494} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 494,\ (\ :1/2),\ 0.999 - 0.0329i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.69287 + 0.0444140i\)
\(L(\frac12)\) \(\approx\) \(2.69287 + 0.0444140i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 + 0.5i)T \)
13 \( 1 + (-3.03 + 1.94i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (-1.32 - 2.29i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 3.69iT - 5T^{2} \)
7 \( 1 + (-3.52 - 2.03i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.94 - 2.85i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.68 + 2.91i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.723 - 1.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.61 + 2.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.87iT - 31T^{2} \)
37 \( 1 + (10.2 - 5.88i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.31 + 0.761i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.525 + 0.910i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.95iT - 47T^{2} \)
53 \( 1 + 7.94T + 53T^{2} \)
59 \( 1 + (1.57 + 0.906i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.91 + 11.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.7 - 6.18i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (6.40 + 3.69i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.81iT - 73T^{2} \)
79 \( 1 - 4.47T + 79T^{2} \)
83 \( 1 + 7.57iT - 83T^{2} \)
89 \( 1 + (-4.25 + 2.45i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.4 + 6.00i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.86749887568192941662260714026, −10.07233474288148276009424085688, −9.143929158222664438972886136428, −8.461864369951954712030794426291, −7.81634632265737358678494475159, −5.45890250412185387073975038173, −5.00843973911995435095798608422, −4.50459511594656266590171520672, −3.11176865390048636639862247672, −1.76330102339889887280947479877, 1.81837903110840206062379676014, 2.87694127275743024588036958028, 3.87888511174833665488283338891, 5.57803096946363502269780200429, 6.55387075954045963151630472910, 7.38577974503079505315957186704, 7.87220735318838619571842531896, 8.566294666078470171883776480208, 10.63730913572039696776189657055, 10.83755836557814309122574176065

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