↑

Properties

Label 2-494-13.10-c1-0-9
Degree $2$
Conductor $494$
Sign $0.668 + 0.743i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (−0.792 − 1.37i)3-s + (0.499 − 0.866i)4-s + 2.09i·5-s + (−1.37 − 0.792i)6-s + (2.83 + 1.63i)7-s − 0.999i·8-s + (0.243 − 0.421i)9-s + (1.04 + 1.81i)10-s + (0.161 − 0.0934i)11-s − 1.58·12-s + (3.44 + 1.07i)13-s + 3.27·14-s + (2.87 − 1.65i)15-s + (−0.5 − 0.866i)16-s + (1.41 − 2.44i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.457 − 0.792i)3-s + (0.249 − 0.433i)4-s + 0.934i·5-s + (−0.560 − 0.323i)6-s + (1.07 + 0.618i)7-s − 0.353i·8-s + (0.0810 − 0.140i)9-s + (0.330 + 0.572i)10-s + (0.0488 − 0.0281i)11-s − 0.457·12-s + (0.954 + 0.297i)13-s + 0.875·14-s + (0.741 − 0.427i)15-s + (−0.125 − 0.216i)16-s + (0.343 − 0.594i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 494 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.668 + 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(494\)    =    \(2 \cdot 13 \cdot 19\)
Sign: $0.668 + 0.743i$
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: \((2,\ 494,\ (\ :1/2),\ 0.668 + 0.743i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82979 - 0.814959i\)
\(L(\frac12)\) \(\approx\) \(1.82979 - 0.814959i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.44 - 1.07i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (0.792 + 1.37i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 2.09iT - 5T^{2} \)
7 \( 1 + (-2.83 - 1.63i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.161 + 0.0934i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.41 + 2.44i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.766 + 1.32i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.51 + 4.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.59iT - 31T^{2} \)
37 \( 1 + (-1.78 + 1.03i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.08 + 1.78i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.84 - 6.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.47iT - 47T^{2} \)
53 \( 1 - 1.39T + 53T^{2} \)
59 \( 1 + (1.68 + 0.970i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.15 - 7.20i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.01 - 0.586i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.68 + 1.54i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.9iT - 73T^{2} \)
79 \( 1 + 2.06T + 79T^{2} \)
83 \( 1 - 13.1iT - 83T^{2} \)
89 \( 1 + (-7.96 + 4.60i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.637 - 0.367i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.26656141038414637625512947189, −10.28749070334886717477843818252, −9.058225495366153024522073806883, −7.917688570772630713432047345175, −6.91642601614644012196793305172, −6.20392616011389278401072611504, −5.27688756812313837538541392741, −3.97505896099134376998588457389, −2.63304710758369728750841317362, −1.41200303461300241333834715813, 1.52025133213904128939553208453, 3.71760742116380038415588401275, 4.51735045932222442361014893794, 5.19814538263773733689392198099, 6.08204241230113567631324369597, 7.56619792395015599436252051631, 8.229989420144404657425328644387, 9.227145089620927522921947822990, 10.49553806991273319196519374878, 10.99355848808481653063541845498

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