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Properties

Label 2-494-13.10-c1-0-12
Degree $2$
Conductor $494$
Sign $0.936 + 0.350i$
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 + (0.449 + 0.778i)3-s + (0.499 − 0.866i)4-s − 0.684i·5-s + (0.778 + 0.449i)6-s + (2.72 + 1.57i)7-s − 0.999i·8-s + (1.09 − 1.89i)9-s + (−0.342 − 0.592i)10-s + (1.21 − 0.699i)11-s + 0.898·12-s + (−3.54 + 0.639i)13-s + 3.14·14-s + (0.532 − 0.307i)15-s + (−0.5 − 0.866i)16-s + (−0.975 + 1.69i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.259 + 0.449i)3-s + (0.249 − 0.433i)4-s − 0.305i·5-s + (0.317 + 0.183i)6-s + (1.02 + 0.593i)7-s − 0.353i·8-s + (0.365 − 0.632i)9-s + (−0.108 − 0.187i)10-s + (0.365 − 0.210i)11-s + 0.259·12-s + (−0.984 + 0.177i)13-s + 0.839·14-s + (0.137 − 0.0793i)15-s + (−0.125 − 0.216i)16-s + (−0.236 + 0.409i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.350i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(494\)    =    \(2 \cdot 13 \cdot 19\)
Sign: $0.936 + 0.350i$
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: \((2,\ 494,\ (\ :1/2),\ 0.936 + 0.350i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35641 - 0.426456i\)
\(L(\frac12)\) \(\approx\) \(2.35641 - 0.426456i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.54 - 0.639i)T \)
19 \( 1 + (-0.866 - 0.5i)T \)
good3 \( 1 + (-0.449 - 0.778i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + 0.684iT - 5T^{2} \)
7 \( 1 + (-2.72 - 1.57i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.21 + 0.699i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (0.975 - 1.69i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.10 - 1.91i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.92 + 3.33i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.71iT - 31T^{2} \)
37 \( 1 + (-2.19 + 1.26i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.58 - 3.22i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.32 - 2.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 1.10iT - 47T^{2} \)
53 \( 1 + 9.51T + 53T^{2} \)
59 \( 1 + (3.82 + 2.20i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.15 - 10.6i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.12 - 3.53i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (7.16 + 4.13i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.1iT - 73T^{2} \)
79 \( 1 + 5.19T + 79T^{2} \)
83 \( 1 - 1.27iT - 83T^{2} \)
89 \( 1 + (0.584 - 0.337i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-4.92 - 2.84i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.09003613679890467026571321948, −9.982898917389025029364993135270, −9.239738544446745381361267075693, −8.377899619441910043473618619559, −7.19466681606380490720118296791, −6.01556291868184777901116282738, −4.92920708288770905079123577911, −4.25501590392709643323051349453, −2.96861081879866518016243532327, −1.56451583010449087033145632655, 1.72330723836916521062140676433, 3.01396691911002030953947623909, 4.57181829930293382618595885316, 5.04120272467649452988847536012, 6.62908243508328200210277684862, 7.34714169306001080713857051737, 7.910474766012065491175362313007, 9.034213460774194577056753605121, 10.36040633319124521864695212639, 11.02662531165382403091303463731

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