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Properties

Label 2-40-8.5-c3-0-3
Degree $2$
Conductor $40$
Sign $-0.137 - 0.990i$
Analytic cond. $2.36007$
Root an. cond. $1.53625$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.52 + 2.38i)2-s + 1.51i·3-s + (−3.34 + 7.26i)4-s + 5i·5-s + (−3.61 + 2.31i)6-s + 5.13·7-s + (−22.4 + 3.11i)8-s + 24.6·9-s + (−11.9 + 7.62i)10-s − 31.3i·11-s + (−11.0 − 5.08i)12-s + 4.75i·13-s + (7.83 + 12.2i)14-s − 7.58·15-s + (−41.5 − 48.6i)16-s + 108.·17-s + ⋯
L(s)  = 1  + (0.539 + 0.842i)2-s + 0.292i·3-s + (−0.418 + 0.908i)4-s + 0.447i·5-s + (−0.245 + 0.157i)6-s + 0.277·7-s + (−0.990 + 0.137i)8-s + 0.914·9-s + (−0.376 + 0.241i)10-s − 0.859i·11-s + (−0.265 − 0.122i)12-s + 0.101i·13-s + (0.149 + 0.233i)14-s − 0.130·15-s + (−0.649 − 0.760i)16-s + 1.54·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.137 - 0.990i$
Analytic conductor: \(2.36007\)
Root analytic conductor: \(1.53625\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :3/2),\ -0.137 - 0.990i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.07987 + 1.24010i\)
\(L(\frac12)\) \(\approx\) \(1.07987 + 1.24010i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.52 - 2.38i)T \)
5 \( 1 - 5iT \)
good3 \( 1 - 1.51iT - 27T^{2} \)
7 \( 1 - 5.13T + 343T^{2} \)
11 \( 1 + 31.3iT - 1.33e3T^{2} \)
13 \( 1 - 4.75iT - 2.19e3T^{2} \)
17 \( 1 - 108.T + 4.91e3T^{2} \)
19 \( 1 + 89.8iT - 6.85e3T^{2} \)
23 \( 1 + 68.5T + 1.21e4T^{2} \)
29 \( 1 - 16.5iT - 2.43e4T^{2} \)
31 \( 1 + 300.T + 2.97e4T^{2} \)
37 \( 1 - 327. iT - 5.06e4T^{2} \)
41 \( 1 + 73.4T + 6.89e4T^{2} \)
43 \( 1 + 0.836iT - 7.95e4T^{2} \)
47 \( 1 - 228.T + 1.03e5T^{2} \)
53 \( 1 + 647. iT - 1.48e5T^{2} \)
59 \( 1 + 753. iT - 2.05e5T^{2} \)
61 \( 1 + 290. iT - 2.26e5T^{2} \)
67 \( 1 - 801. iT - 3.00e5T^{2} \)
71 \( 1 - 767.T + 3.57e5T^{2} \)
73 \( 1 + 48.3T + 3.89e5T^{2} \)
79 \( 1 - 451.T + 4.93e5T^{2} \)
83 \( 1 - 976. iT - 5.71e5T^{2} \)
89 \( 1 + 1.20e3T + 7.04e5T^{2} \)
97 \( 1 + 559.T + 9.12e5T^{2} \)
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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

−15.90919812438389449606523396793, −14.85065445264410074848362896097, −13.89937617118085380842780986125, −12.70374236187986572551256307782, −11.29539748479588318534863805321, −9.694209435180854093597001602352, −8.108243066520563743373122054631, −6.82768193421452688484910150784, −5.25669062949943820978526160606, −3.56691018322110164998638809421, 1.61151751913334121692946290309, 4.03630318301966010731017565105, 5.59604141827610957809989191325, 7.60644093569104955898231651394, 9.483748412303945676521570273468, 10.49538047982462241736438831847, 12.19427482081062540598154264328, 12.62426575775220699460949621690, 14.01324630314149553972858772960, 15.04160841572042493495571284812

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