Properties

Label 2-40-40.13-c10-0-8
Degree $2$
Conductor $40$
Sign $0.261 + 0.965i$
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  + (−17.2 + 26.9i)2-s + (−31.2 + 31.2i)3-s + (−431. − 928. i)4-s + (−1.81e3 + 2.54e3i)5-s + (−305. − 1.38e3i)6-s + (−1.32e4 + 1.32e4i)7-s + (3.24e4 + 4.33e3i)8-s + 5.70e4i·9-s + (−3.73e4 − 9.27e4i)10-s + 1.50e5i·11-s + (4.25e4 + 1.55e4i)12-s + (−3.35e4 + 3.35e4i)13-s + (−1.29e5 − 5.85e5i)14-s + (−2.27e4 − 1.36e5i)15-s + (−6.76e5 + 8.01e5i)16-s + (−4.88e5 + 4.88e5i)17-s + ⋯
L(s)  = 1  + (−0.537 + 0.843i)2-s + (−0.128 + 0.128i)3-s + (−0.421 − 0.906i)4-s + (−0.580 + 0.813i)5-s + (−0.0392 − 0.177i)6-s + (−0.788 + 0.788i)7-s + (0.991 + 0.132i)8-s + 0.966i·9-s + (−0.373 − 0.927i)10-s + 0.937i·11-s + (0.170 + 0.0624i)12-s + (−0.0903 + 0.0903i)13-s + (−0.240 − 1.08i)14-s + (−0.0299 − 0.179i)15-s + (−0.644 + 0.764i)16-s + (−0.343 + 0.343i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.261 + 0.965i$
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: \((2,\ 40,\ (\ :5),\ 0.261 + 0.965i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.258763 - 0.197995i\)
\(L(\frac12)\) \(\approx\) \(0.258763 - 0.197995i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (17.2 - 26.9i)T \)
5 \( 1 + (1.81e3 - 2.54e3i)T \)
good3 \( 1 + (31.2 - 31.2i)T - 5.90e4iT^{2} \)
7 \( 1 + (1.32e4 - 1.32e4i)T - 2.82e8iT^{2} \)
11 \( 1 - 1.50e5iT - 2.59e10T^{2} \)
13 \( 1 + (3.35e4 - 3.35e4i)T - 1.37e11iT^{2} \)
17 \( 1 + (4.88e5 - 4.88e5i)T - 2.01e12iT^{2} \)
19 \( 1 + 2.78e6T + 6.13e12T^{2} \)
23 \( 1 + (-1.61e6 - 1.61e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 1.47e7T + 4.20e14T^{2} \)
31 \( 1 - 4.06e7T + 8.19e14T^{2} \)
37 \( 1 + (7.22e7 + 7.22e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 1.28e8T + 1.34e16T^{2} \)
43 \( 1 + (2.06e7 - 2.06e7i)T - 2.16e16iT^{2} \)
47 \( 1 + (-5.20e7 + 5.20e7i)T - 5.25e16iT^{2} \)
53 \( 1 + (-2.51e7 + 2.51e7i)T - 1.74e17iT^{2} \)
59 \( 1 + 9.05e8T + 5.11e17T^{2} \)
61 \( 1 - 6.35e8iT - 7.13e17T^{2} \)
67 \( 1 + (9.08e8 + 9.08e8i)T + 1.82e18iT^{2} \)
71 \( 1 - 1.73e9T + 3.25e18T^{2} \)
73 \( 1 + (-1.36e9 - 1.36e9i)T + 4.29e18iT^{2} \)
79 \( 1 - 1.93e9iT - 9.46e18T^{2} \)
83 \( 1 + (4.07e9 - 4.07e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 1.03e10iT - 3.11e19T^{2} \)
97 \( 1 + (2.48e9 - 2.48e9i)T - 7.37e19iT^{2} \)
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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

−15.29018295248698094697146023795, −14.03533150763150702696150356608, −12.54381852126298859346597851239, −10.90615501228370987203356160522, −9.934397742288247524732972583055, −8.506832083709502232902645897341, −7.23381570556423366657023664321, −6.13320978925589829082016889811, −4.50169741722030286023370778011, −2.31741175714108403405937643385, 0.17210621635703781410480028387, 0.940539652530838893767950324161, 3.19429261537821123610933453426, 4.36220770802768467338023707384, 6.65068088734179722073778922763, 8.240608117701826749379037179466, 9.257574704245790213742758795196, 10.56196555423854738407684421190, 11.81944128539987462351511398882, 12.72406734982172838976576041429

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