Properties

Label 2-40-40.13-c10-0-42
Degree $2$
Conductor $40$
Sign $0.0766 + 0.997i$
Analytic cond. $25.4142$
Root an. cond. $5.04125$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−31.8 − 2.70i)2-s + (−181. + 181. i)3-s + (1.00e3 + 172. i)4-s + (1.98e3 − 2.41e3i)5-s + (6.27e3 − 5.29e3i)6-s + (1.60e4 − 1.60e4i)7-s + (−3.17e4 − 8.23e3i)8-s − 6.77e3i·9-s + (−6.96e4 + 7.17e4i)10-s + 9.36e3i·11-s + (−2.14e5 + 1.51e5i)12-s + (−4.47e4 + 4.47e4i)13-s + (−5.55e5 + 4.68e5i)14-s + (7.93e4 + 7.97e5i)15-s + (9.88e5 + 3.48e5i)16-s + (4.09e5 − 4.09e5i)17-s + ⋯
L(s)  = 1  + (−0.996 − 0.0846i)2-s + (−0.746 + 0.746i)3-s + (0.985 + 0.168i)4-s + (0.633 − 0.773i)5-s + (0.807 − 0.680i)6-s + (0.955 − 0.955i)7-s + (−0.967 − 0.251i)8-s − 0.114i·9-s + (−0.696 + 0.717i)10-s + 0.0581i·11-s + (−0.861 + 0.609i)12-s + (−0.120 + 0.120i)13-s + (−1.03 + 0.871i)14-s + (0.104 + 1.05i)15-s + (0.943 + 0.332i)16-s + (0.288 − 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0766 + 0.997i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.0766 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.0766 + 0.997i$
Analytic conductor: \(25.4142\)
Root analytic conductor: \(5.04125\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5),\ 0.0766 + 0.997i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.626598 - 0.580267i\)
\(L(\frac12)\) \(\approx\) \(0.626598 - 0.580267i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (31.8 + 2.70i)T \)
5 \( 1 + (-1.98e3 + 2.41e3i)T \)
good3 \( 1 + (181. - 181. i)T - 5.90e4iT^{2} \)
7 \( 1 + (-1.60e4 + 1.60e4i)T - 2.82e8iT^{2} \)
11 \( 1 - 9.36e3iT - 2.59e10T^{2} \)
13 \( 1 + (4.47e4 - 4.47e4i)T - 1.37e11iT^{2} \)
17 \( 1 + (-4.09e5 + 4.09e5i)T - 2.01e12iT^{2} \)
19 \( 1 + 2.36e6T + 6.13e12T^{2} \)
23 \( 1 + (-2.24e6 - 2.24e6i)T + 4.14e13iT^{2} \)
29 \( 1 + 2.96e7T + 4.20e14T^{2} \)
31 \( 1 - 3.83e7T + 8.19e14T^{2} \)
37 \( 1 + (1.26e7 + 1.26e7i)T + 4.80e15iT^{2} \)
41 \( 1 + 7.38e7T + 1.34e16T^{2} \)
43 \( 1 + (-1.96e8 + 1.96e8i)T - 2.16e16iT^{2} \)
47 \( 1 + (-2.72e8 + 2.72e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (-2.93e7 + 2.93e7i)T - 1.74e17iT^{2} \)
59 \( 1 + 6.79e8T + 5.11e17T^{2} \)
61 \( 1 - 1.01e9iT - 7.13e17T^{2} \)
67 \( 1 + (3.41e8 + 3.41e8i)T + 1.82e18iT^{2} \)
71 \( 1 + 6.24e8T + 3.25e18T^{2} \)
73 \( 1 + (2.00e9 + 2.00e9i)T + 4.29e18iT^{2} \)
79 \( 1 + 5.65e9iT - 9.46e18T^{2} \)
83 \( 1 + (-1.18e9 + 1.18e9i)T - 1.55e19iT^{2} \)
89 \( 1 - 1.86e9iT - 3.11e19T^{2} \)
97 \( 1 + (-4.01e9 + 4.01e9i)T - 7.37e19iT^{2} \)
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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

−13.63369904993340600217630022960, −11.99965212028586658652696919626, −10.84774900388031088049287398403, −10.13117220086348885436265835386, −8.838313420722805323210816796287, −7.46679290513208744811665523490, −5.69794070918525861088272324509, −4.37958389448043086699023177400, −1.82198932644689945690783078958, −0.45515433779101537480944400690, 1.32099850941277334007285534683, 2.47550813999457741660631422970, 5.65664776296383338591316404044, 6.51970616823194150430359064704, 7.84243550185943393116478900786, 9.228050124753796723884605731460, 10.71321212068736394386980279895, 11.55301697190859351153345219081, 12.64678600203290167492249020622, 14.52629714711664263913704589328

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