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Properties

Label 2-494-13.10-c1-0-21
Degree $2$
Conductor $494$
Sign $-0.351 - 0.936i$
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 + (−1.68 − 2.91i)3-s + (0.499 − 0.866i)4-s + 3.01i·5-s + (−2.91 − 1.68i)6-s + (−1.39 − 0.802i)7-s − 0.999i·8-s + (−4.15 + 7.20i)9-s + (1.50 + 2.61i)10-s + (−4.10 + 2.37i)11-s − 3.36·12-s + (−2.81 − 2.25i)13-s − 1.60·14-s + (8.79 − 5.07i)15-s + (−0.5 − 0.866i)16-s + (−1.57 + 2.72i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.971 − 1.68i)3-s + (0.249 − 0.433i)4-s + 1.34i·5-s + (−1.18 − 0.686i)6-s + (−0.525 − 0.303i)7-s − 0.353i·8-s + (−1.38 + 2.40i)9-s + (0.477 + 0.826i)10-s + (−1.23 + 0.715i)11-s − 0.971·12-s + (−0.780 − 0.625i)13-s − 0.429·14-s + (2.27 − 1.31i)15-s + (−0.125 − 0.216i)16-s + (−0.382 + 0.661i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.351 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(494\)    =    \(2 \cdot 13 \cdot 19\)
Sign: $-0.351 - 0.936i$
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: \((2,\ 494,\ (\ :1/2),\ -0.351 - 0.936i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0391628 + 0.0565186i\)
\(L(\frac12)\) \(\approx\) \(0.0391628 + 0.0565186i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.81 + 2.25i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (1.68 + 2.91i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3.01iT - 5T^{2} \)
7 \( 1 + (1.39 + 0.802i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (4.10 - 2.37i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.57 - 2.72i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.21 + 3.84i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.991 + 1.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.75iT - 31T^{2} \)
37 \( 1 + (-7.31 + 4.22i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.94 - 3.43i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.19 + 3.80i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 12.7iT - 47T^{2} \)
53 \( 1 + 9.81T + 53T^{2} \)
59 \( 1 + (4.81 + 2.78i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.50 - 6.07i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.73 + 2.15i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.443 + 0.256i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 0.764iT - 73T^{2} \)
79 \( 1 + 1.84T + 79T^{2} \)
83 \( 1 + 15.5iT - 83T^{2} \)
89 \( 1 + (-5.93 + 3.42i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.854 - 0.493i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.67637898846680991798748963109, −10.02455000335084707294310612123, −7.86791939121365137925045389297, −7.40918014598486365829374370194, −6.47730673232365154582339700048, −5.96586546109596199177240987577, −4.74651222808726175848560022157, −2.83756415015352539695830159051, −2.15838237894786753477284549130, −0.03461462059287344876768309995, 3.07999219058118768645716429995, 4.27906488884107900223815205700, 5.08429525294544071967183511527, 5.46229573731139893705842703356, 6.53415557859392375476525287585, 8.162917949411130634462055473824, 9.148847219346782192036284888263, 9.699956694184792896903861871905, 10.73892853690130601549460691503, 11.63405476120631718941835763998

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