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Properties

Label 2-40-40.13-c2-0-8
Degree $2$
Conductor $40$
Sign $0.334 + 0.942i$
Analytic cond. $1.08992$
Root an. cond. $1.04399$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.764 − 1.84i)2-s + (−0.130 + 0.130i)3-s + (−2.83 − 2.82i)4-s + (4.38 − 2.39i)5-s + (0.141 + 0.341i)6-s + (−1.59 + 1.59i)7-s + (−7.38 + 3.07i)8-s + 8.96i·9-s + (−1.07 − 9.94i)10-s + 11.9i·11-s + (0.739 − 0.000715i)12-s + (9.59 − 9.59i)13-s + (1.73 + 4.17i)14-s + (−0.260 + 0.887i)15-s + (0.0309 + 15.9i)16-s + (0.857 − 0.857i)17-s + ⋯
L(s)  = 1  + (0.382 − 0.924i)2-s + (−0.0435 + 0.0435i)3-s + (−0.707 − 0.706i)4-s + (0.877 − 0.479i)5-s + (0.0236 + 0.0569i)6-s + (−0.228 + 0.228i)7-s + (−0.923 + 0.384i)8-s + 0.996i·9-s + (−0.107 − 0.994i)10-s + 1.08i·11-s + (0.0616 − 5.96e−5i)12-s + (0.738 − 0.738i)13-s + (0.123 + 0.298i)14-s + (−0.0173 + 0.0591i)15-s + (0.00193 + 0.999i)16-s + (0.0504 − 0.0504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.334 + 0.942i$
Analytic conductor: \(1.08992\)
Root analytic conductor: \(1.04399\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :1),\ 0.334 + 0.942i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.02481 - 0.723894i\)
\(L(\frac12)\) \(\approx\) \(1.02481 - 0.723894i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.764 + 1.84i)T \)
5 \( 1 + (-4.38 + 2.39i)T \)
good3 \( 1 + (0.130 - 0.130i)T - 9iT^{2} \)
7 \( 1 + (1.59 - 1.59i)T - 49iT^{2} \)
11 \( 1 - 11.9iT - 121T^{2} \)
13 \( 1 + (-9.59 + 9.59i)T - 169iT^{2} \)
17 \( 1 + (-0.857 + 0.857i)T - 289iT^{2} \)
19 \( 1 + 20.5T + 361T^{2} \)
23 \( 1 + (22.1 + 22.1i)T + 529iT^{2} \)
29 \( 1 + 27.3T + 841T^{2} \)
31 \( 1 - 40.0T + 961T^{2} \)
37 \( 1 + (1.57 + 1.57i)T + 1.36e3iT^{2} \)
41 \( 1 + 37.5T + 1.68e3T^{2} \)
43 \( 1 + (-49.2 + 49.2i)T - 1.84e3iT^{2} \)
47 \( 1 + (34.0 - 34.0i)T - 2.20e3iT^{2} \)
53 \( 1 + (28.8 - 28.8i)T - 2.80e3iT^{2} \)
59 \( 1 - 92.7T + 3.48e3T^{2} \)
61 \( 1 + 4.82iT - 3.72e3T^{2} \)
67 \( 1 + (54.6 + 54.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 59.2T + 5.04e3T^{2} \)
73 \( 1 + (-34.1 - 34.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 96.2iT - 6.24e3T^{2} \)
83 \( 1 + (63.6 - 63.6i)T - 6.88e3iT^{2} \)
89 \( 1 + 3.68iT - 7.92e3T^{2} \)
97 \( 1 + (-46.0 + 46.0i)T - 9.40e3iT^{2} \)
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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

−15.56023172695676305770310837546, −14.18988066028764433539213372045, −13.14363567830613533989710181117, −12.39992109809274126303292069302, −10.71783751886155069485603915039, −9.880468774796374623934585675903, −8.467971962474587957291425221452, −5.98032687136995174376314663302, −4.60535356353336603120044404360, −2.17195988595542199136343272521, 3.66439460205249472334318488960, 5.91209662360198874010712963564, 6.67219267800265732205754429287, 8.542420080536914006499518088814, 9.752797556818390506871719664388, 11.50938703097887808951199414735, 13.11045069474501504299133055615, 13.91054606355536525518339414497, 14.91717294191074201435213068076, 16.15782466273341269618485461370

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