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Properties

Label 2-40-5.4-c5-0-5
Degree $2$
Conductor $40$
Sign $-0.317 + 0.948i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.48i·3-s + (−53.0 − 17.7i)5-s − 188. i·7-s + 212.·9-s − 501.·11-s − 1.06e3i·13-s + (97.5 − 290. i)15-s − 29.5i·17-s − 1.57e3·19-s + 1.03e3·21-s + 1.29e3i·23-s + (2.49e3 + 1.88e3i)25-s + 2.50e3i·27-s + 3.58e3·29-s − 3.52e3·31-s + ⋯
L(s)  = 1  + 0.352i·3-s + (−0.948 − 0.317i)5-s − 1.45i·7-s + 0.875·9-s − 1.25·11-s − 1.74i·13-s + (0.111 − 0.333i)15-s − 0.0248i·17-s − 1.00·19-s + 0.513·21-s + 0.510i·23-s + (0.798 + 0.602i)25-s + 0.660i·27-s + 0.791·29-s − 0.659·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.317 + 0.948i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ -0.317 + 0.948i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.543276 - 0.755010i\)
\(L(\frac12)\) \(\approx\) \(0.543276 - 0.755010i\)
\(L(\frac{7}{2})\) 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 \( 1 + (53.0 + 17.7i)T \)
good3 \( 1 - 5.48iT - 243T^{2} \)
7 \( 1 + 188. iT - 1.68e4T^{2} \)
11 \( 1 + 501.T + 1.61e5T^{2} \)
13 \( 1 + 1.06e3iT - 3.71e5T^{2} \)
17 \( 1 + 29.5iT - 1.41e6T^{2} \)
19 \( 1 + 1.57e3T + 2.47e6T^{2} \)
23 \( 1 - 1.29e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.58e3T + 2.05e7T^{2} \)
31 \( 1 + 3.52e3T + 2.86e7T^{2} \)
37 \( 1 - 8.41e3iT - 6.93e7T^{2} \)
41 \( 1 - 7.01e3T + 1.15e8T^{2} \)
43 \( 1 + 2.26e4iT - 1.47e8T^{2} \)
47 \( 1 + 3.50e3iT - 2.29e8T^{2} \)
53 \( 1 + 2.73e4iT - 4.18e8T^{2} \)
59 \( 1 + 7.92e3T + 7.14e8T^{2} \)
61 \( 1 + 7.02e3T + 8.44e8T^{2} \)
67 \( 1 - 1.76e4iT - 1.35e9T^{2} \)
71 \( 1 - 1.34e4T + 1.80e9T^{2} \)
73 \( 1 + 3.99e4iT - 2.07e9T^{2} \)
79 \( 1 - 9.33e4T + 3.07e9T^{2} \)
83 \( 1 - 5.84e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.39e4T + 5.58e9T^{2} \)
97 \( 1 + 1.10e5iT - 8.58e9T^{2} \)
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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

−15.15220684062994629414658337110, −13.37547702123364545523126203596, −12.62673959235596732142378157298, −10.77128188657528469013287822515, −10.22718531971566761434107279348, −8.147927579946218317637028837575, −7.27424335180375772775752086208, −4.92942033400867577619390194707, −3.61760718192523743863661783790, −0.51859170610340627233226777625, 2.34743002647911654604635093166, 4.50634306826845532649400901187, 6.45016915856492289261098685550, 7.82593728203879241133885593083, 9.103679168441523076114468692282, 10.82246251809712248764333879601, 12.05812820030051063967523627761, 12.84036440894720971207828450473, 14.51464696224036349427594845355, 15.58064991072424349914974203236

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