Properties

Label 2-40-40.13-c10-0-24
Degree $2$
Conductor $40$
Sign $0.999 - 0.00445i$
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  + (−30.5 − 9.47i)2-s + (41.2 − 41.2i)3-s + (844. + 579. i)4-s + (−3.10e3 − 346. i)5-s + (−1.65e3 + 870. i)6-s + (4.45e3 − 4.45e3i)7-s + (−2.03e4 − 2.56e4i)8-s + 5.56e4i·9-s + (9.16e4 + 4.00e4i)10-s − 2.04e5i·11-s + (5.87e4 − 1.09e4i)12-s + (−9.10e4 + 9.10e4i)13-s + (−1.78e5 + 9.39e4i)14-s + (−1.42e5 + 1.13e5i)15-s + (3.77e5 + 9.78e5i)16-s + (−1.56e6 + 1.56e6i)17-s + ⋯
L(s)  = 1  + (−0.955 − 0.296i)2-s + (0.169 − 0.169i)3-s + (0.824 + 0.565i)4-s + (−0.993 − 0.110i)5-s + (−0.212 + 0.111i)6-s + (0.265 − 0.265i)7-s + (−0.620 − 0.784i)8-s + 0.942i·9-s + (0.916 + 0.400i)10-s − 1.26i·11-s + (0.236 − 0.0440i)12-s + (−0.245 + 0.245i)13-s + (−0.331 + 0.174i)14-s + (−0.187 + 0.149i)15-s + (0.360 + 0.932i)16-s + (−1.10 + 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00445i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.999 - 0.00445i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.999 - 0.00445i$
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: \((2,\ 40,\ (\ :5),\ 0.999 - 0.00445i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.919822 + 0.00204868i\)
\(L(\frac12)\) \(\approx\) \(0.919822 + 0.00204868i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (30.5 + 9.47i)T \)
5 \( 1 + (3.10e3 + 346. i)T \)
good3 \( 1 + (-41.2 + 41.2i)T - 5.90e4iT^{2} \)
7 \( 1 + (-4.45e3 + 4.45e3i)T - 2.82e8iT^{2} \)
11 \( 1 + 2.04e5iT - 2.59e10T^{2} \)
13 \( 1 + (9.10e4 - 9.10e4i)T - 1.37e11iT^{2} \)
17 \( 1 + (1.56e6 - 1.56e6i)T - 2.01e12iT^{2} \)
19 \( 1 + 7.59e5T + 6.13e12T^{2} \)
23 \( 1 + (4.30e6 + 4.30e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 3.88e7T + 4.20e14T^{2} \)
31 \( 1 - 2.62e7T + 8.19e14T^{2} \)
37 \( 1 + (-6.23e7 - 6.23e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 6.75e6T + 1.34e16T^{2} \)
43 \( 1 + (-1.50e8 + 1.50e8i)T - 2.16e16iT^{2} \)
47 \( 1 + (-8.45e7 + 8.45e7i)T - 5.25e16iT^{2} \)
53 \( 1 + (-2.25e6 + 2.25e6i)T - 1.74e17iT^{2} \)
59 \( 1 - 6.90e8T + 5.11e17T^{2} \)
61 \( 1 + 2.62e7iT - 7.13e17T^{2} \)
67 \( 1 + (1.06e9 + 1.06e9i)T + 1.82e18iT^{2} \)
71 \( 1 - 2.58e9T + 3.25e18T^{2} \)
73 \( 1 + (-2.04e9 - 2.04e9i)T + 4.29e18iT^{2} \)
79 \( 1 - 3.06e9iT - 9.46e18T^{2} \)
83 \( 1 + (1.92e9 - 1.92e9i)T - 1.55e19iT^{2} \)
89 \( 1 - 4.48e9iT - 3.11e19T^{2} \)
97 \( 1 + (6.51e7 - 6.51e7i)T - 7.37e19iT^{2} \)
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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

−13.85903034094413645753235008982, −12.45058884487160600910790423739, −11.23342615340326352830130674460, −10.49635273205220528623853903190, −8.515249425563944005193072964452, −8.098269690578940085271773063384, −6.57581267957066787121283971763, −4.21661664751775167866274807440, −2.53702509682023982909677683438, −0.811907370627714802127917147422, 0.61414498300175731052309025137, 2.57076064330494036489137236465, 4.54090583330349059966577220796, 6.59350366980043643072985664943, 7.67782177612880640094274687618, 8.918090177225031641634993567222, 10.00930382475215687505372421393, 11.47204248445864298451160245935, 12.29239743320549168346723509352, 14.47479671831502196196808023212

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