↑

Properties

Label 2-40-40.27-c5-0-23
Degree $2$
Conductor $40$
Sign $0.559 + 0.828i$
Analytic cond. $6.41535$
Root an. cond. $2.53285$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (5.57 − 0.946i)2-s + (4.45 − 4.45i)3-s + (30.2 − 10.5i)4-s + (−49.9 − 25.0i)5-s + (20.6 − 29.0i)6-s + (169. − 169. i)7-s + (158. − 87.4i)8-s + 203. i·9-s + (−302. − 92.2i)10-s − 207.·11-s + (87.5 − 181. i)12-s + (249. + 249. i)13-s + (783. − 1.10e3i)14-s + (−334. + 111. i)15-s + (801. − 637. i)16-s + (−424. − 424. i)17-s + ⋯
L(s)  = 1  + (0.985 − 0.167i)2-s + (0.285 − 0.285i)3-s + (0.944 − 0.329i)4-s + (−0.894 − 0.447i)5-s + (0.233 − 0.329i)6-s + (1.30 − 1.30i)7-s + (0.875 − 0.483i)8-s + 0.836i·9-s + (−0.956 − 0.291i)10-s − 0.515·11-s + (0.175 − 0.364i)12-s + (0.408 + 0.408i)13-s + (1.06 − 1.50i)14-s + (−0.383 + 0.127i)15-s + (0.782 − 0.622i)16-s + (−0.355 − 0.355i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.559 + 0.828i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.559 + 0.828i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ 0.559 + 0.828i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.62258 - 1.39395i\)
\(L(\frac12)\) \(\approx\) \(2.62258 - 1.39395i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-5.57 + 0.946i)T \)
5 \( 1 + (49.9 + 25.0i)T \)
good3 \( 1 + (-4.45 + 4.45i)T - 243iT^{2} \)
7 \( 1 + (-169. + 169. i)T - 1.68e4iT^{2} \)
11 \( 1 + 207.T + 1.61e5T^{2} \)
13 \( 1 + (-249. - 249. i)T + 3.71e5iT^{2} \)
17 \( 1 + (424. + 424. i)T + 1.41e6iT^{2} \)
19 \( 1 - 2.28e3iT - 2.47e6T^{2} \)
23 \( 1 + (351. + 351. i)T + 6.43e6iT^{2} \)
29 \( 1 + 7.25e3T + 2.05e7T^{2} \)
31 \( 1 - 7.24e3iT - 2.86e7T^{2} \)
37 \( 1 + (266. - 266. i)T - 6.93e7iT^{2} \)
41 \( 1 - 1.68e4T + 1.15e8T^{2} \)
43 \( 1 + (4.60e3 - 4.60e3i)T - 1.47e8iT^{2} \)
47 \( 1 + (-5.61e3 + 5.61e3i)T - 2.29e8iT^{2} \)
53 \( 1 + (937. + 937. i)T + 4.18e8iT^{2} \)
59 \( 1 + 1.69e4iT - 7.14e8T^{2} \)
61 \( 1 + 2.72e4iT - 8.44e8T^{2} \)
67 \( 1 + (7.57e3 + 7.57e3i)T + 1.35e9iT^{2} \)
71 \( 1 + 1.71e4iT - 1.80e9T^{2} \)
73 \( 1 + (-1.59e4 + 1.59e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 7.24e4T + 3.07e9T^{2} \)
83 \( 1 + (5.86e4 - 5.86e4i)T - 3.93e9iT^{2} \)
89 \( 1 - 3.43e4iT - 5.58e9T^{2} \)
97 \( 1 + (-6.26e4 - 6.26e4i)T + 8.58e9iT^{2} \)
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

−14.58147675017238663898384276406, −13.84461659231076744901690703933, −12.76011195977577158017146398430, −11.36741446878778052438832283536, −10.64715136971566784341972743399, −8.070456550162890548255309887823, −7.34519770073465447505904402166, −5.04263787402696562913048972490, −3.93177060995216305779901294664, −1.56198996217334676014507203465, 2.63221111356935483953169067993, 4.26159629170526680711465550844, 5.77606579773408238438409080608, 7.55438558836342327163575749840, 8.756230391581517099632991782806, 11.05813123834732536978376758436, 11.69460299138459138354728274529, 12.93965358412729115744031015798, 14.58184020858795063125659394795, 15.22228468661593439947387480080

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