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Properties

Label 2-40-40.3-c1-0-3
Degree $2$
Conductor $40$
Sign $-0.156 + 0.987i$
Analytic cond. $0.319401$
Root an. cond. $0.565156$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 − 1.26i)2-s + (−1.61 − 1.61i)3-s + (−1.17 + 1.61i)4-s + (1.90 − 1.17i)5-s + (−1.00 + 3.07i)6-s + (1.17 + 1.17i)7-s + (2.79 + 0.442i)8-s + 2.23i·9-s + (−2.70 − 1.64i)10-s + 1.23·11-s + (4.52 − 0.715i)12-s + (−3.07 + 3.07i)13-s + (0.726 − 2.23i)14-s + (−4.97 − 1.17i)15-s + (−1.23 − 3.80i)16-s + (−1 + i)17-s + ⋯
L(s)  = 1  + (−0.453 − 0.891i)2-s + (−0.934 − 0.934i)3-s + (−0.587 + 0.809i)4-s + (0.850 − 0.525i)5-s + (−0.408 + 1.25i)6-s + (0.444 + 0.444i)7-s + (0.987 + 0.156i)8-s + 0.745i·9-s + (−0.854 − 0.519i)10-s + 0.372·11-s + (1.30 − 0.206i)12-s + (−0.853 + 0.853i)13-s + (0.194 − 0.597i)14-s + (−1.28 − 0.303i)15-s + (−0.309 − 0.951i)16-s + (−0.242 + 0.242i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.156 + 0.987i$
Analytic conductor: \(0.319401\)
Root analytic conductor: \(0.565156\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :1/2),\ -0.156 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.354546 - 0.415121i\)
\(L(\frac12)\) \(\approx\) \(0.354546 - 0.415121i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.642 + 1.26i)T \)
5 \( 1 + (-1.90 + 1.17i)T \)
good3 \( 1 + (1.61 + 1.61i)T + 3iT^{2} \)
7 \( 1 + (-1.17 - 1.17i)T + 7iT^{2} \)
11 \( 1 - 1.23T + 11T^{2} \)
13 \( 1 + (3.07 - 3.07i)T - 13iT^{2} \)
17 \( 1 + (1 - i)T - 17iT^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + (-2.62 + 2.62i)T - 23iT^{2} \)
29 \( 1 + 1.45T + 29T^{2} \)
31 \( 1 - 5.25iT - 31T^{2} \)
37 \( 1 + (-3.07 - 3.07i)T + 37iT^{2} \)
41 \( 1 + 7.70T + 41T^{2} \)
43 \( 1 + (-2.38 - 2.38i)T + 43iT^{2} \)
47 \( 1 + (7.33 + 7.33i)T + 47iT^{2} \)
53 \( 1 + (0.726 - 0.726i)T - 53iT^{2} \)
59 \( 1 + 8.47iT - 59T^{2} \)
61 \( 1 + 9.95iT - 61T^{2} \)
67 \( 1 + (2.38 - 2.38i)T - 67iT^{2} \)
71 \( 1 - 7.05iT - 71T^{2} \)
73 \( 1 + (-8.70 - 8.70i)T + 73iT^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + (4.38 + 4.38i)T + 83iT^{2} \)
89 \( 1 - 6.47iT - 89T^{2} \)
97 \( 1 + (0.236 - 0.236i)T - 97iT^{2} \)
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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

−16.69184901151339180212763658512, −14.33475139257931331986427285574, −13.02491323649065076598292227930, −12.22202925842500432777911679933, −11.35684989566483309777281668456, −9.831183321784146133610365995218, −8.549502753349435868439382998129, −6.75377824996975443850607170105, −5.02815510561311717267520628090, −1.77368710749098304089451956981, 4.78801037182185899462013514513, 5.90558389948330060439191700396, 7.39535373805772663047943239386, 9.385824529663333265018522815854, 10.29335026759830833900227689860, 11.20708139856224911885921759312, 13.36919304082181771780133225909, 14.62373305165097773724706508728, 15.46997704428470619205568813200, 16.82987788734532707385718112032

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