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Properties

Label 2-40-40.29-c5-0-7
Degree $2$
Conductor $40$
Sign $-0.782 - 0.622i$
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  + (−3.25 + 4.62i)2-s + 1.29·3-s + (−10.8 − 30.1i)4-s + (51.3 + 21.9i)5-s + (−4.22 + 6.00i)6-s + 170. i·7-s + (174. + 47.7i)8-s − 241.·9-s + (−268. + 166. i)10-s + 39.8i·11-s + (−14.0 − 39.0i)12-s − 537.·13-s + (−788. − 554. i)14-s + (66.7 + 28.5i)15-s + (−789. + 652. i)16-s + 1.35e3i·17-s + ⋯
L(s)  = 1  + (−0.575 + 0.818i)2-s + 0.0832·3-s + (−0.338 − 0.940i)4-s + (0.919 + 0.393i)5-s + (−0.0478 + 0.0681i)6-s + 1.31i·7-s + (0.964 + 0.264i)8-s − 0.993·9-s + (−0.850 + 0.525i)10-s + 0.0992i·11-s + (−0.0282 − 0.0783i)12-s − 0.881·13-s + (−1.07 − 0.755i)14-s + (0.0765 + 0.0327i)15-s + (−0.770 + 0.637i)16-s + 1.13i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.782 - 0.622i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.782 - 0.622i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ -0.782 - 0.622i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.346669 + 0.993343i\)
\(L(\frac12)\) \(\approx\) \(0.346669 + 0.993343i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.25 - 4.62i)T \)
5 \( 1 + (-51.3 - 21.9i)T \)
good3 \( 1 - 1.29T + 243T^{2} \)
7 \( 1 - 170. iT - 1.68e4T^{2} \)
11 \( 1 - 39.8iT - 1.61e5T^{2} \)
13 \( 1 + 537.T + 3.71e5T^{2} \)
17 \( 1 - 1.35e3iT - 1.41e6T^{2} \)
19 \( 1 - 1.03e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.61e3iT - 6.43e6T^{2} \)
29 \( 1 + 5.91e3iT - 2.05e7T^{2} \)
31 \( 1 - 9.61e3T + 2.86e7T^{2} \)
37 \( 1 + 5.29e3T + 6.93e7T^{2} \)
41 \( 1 + 1.21e4T + 1.15e8T^{2} \)
43 \( 1 - 1.93e4T + 1.47e8T^{2} \)
47 \( 1 + 7.89e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.89e4T + 4.18e8T^{2} \)
59 \( 1 - 3.23e4iT - 7.14e8T^{2} \)
61 \( 1 + 4.88e4iT - 8.44e8T^{2} \)
67 \( 1 + 882.T + 1.35e9T^{2} \)
71 \( 1 + 8.81e3T + 1.80e9T^{2} \)
73 \( 1 + 1.43e4iT - 2.07e9T^{2} \)
79 \( 1 - 4.98e4T + 3.07e9T^{2} \)
83 \( 1 - 1.39e4T + 3.93e9T^{2} \)
89 \( 1 - 7.58e3T + 5.58e9T^{2} \)
97 \( 1 - 6.67e4iT - 8.58e9T^{2} \)
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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

−15.41413731332808047682559419133, −14.71245205661596929315960816702, −13.66157605094747193911703846466, −11.96610525330466444471292593842, −10.31945693807105122053415425432, −9.242146416871298948040846786394, −8.132989137335545588338836788062, −6.28258163245134385482403565515, −5.44237359647823607228864597273, −2.25323905847677450232343710371, 0.69841143869849507903018052437, 2.73051947086627589123682290525, 4.80043977436314623797403756077, 7.08225459439897014377260344687, 8.655009373318105006715506802051, 9.808278190041528677583639614641, 10.81584157546044334617635488406, 12.16302415492334674335357809015, 13.53087542806817362088007958625, 14.14833964198918175221648437540

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