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Properties

Label 2-40-8.5-c1-0-1
Degree $2$
Conductor $40$
Sign $0.707 - 0.707i$
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  + (0.366 + 1.36i)2-s − 0.732i·3-s + (−1.73 + i)4-s − i·5-s + (1 − 0.267i)6-s − 2.73·7-s + (−2 − 1.99i)8-s + 2.46·9-s + (1.36 − 0.366i)10-s − 2i·11-s + (0.732 + 1.26i)12-s + 3.46i·13-s + (−1 − 3.73i)14-s − 0.732·15-s + (1.99 − 3.46i)16-s − 3.46·17-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s − 0.422i·3-s + (−0.866 + 0.5i)4-s − 0.447i·5-s + (0.408 − 0.109i)6-s − 1.03·7-s + (−0.707 − 0.707i)8-s + 0.821·9-s + (0.431 − 0.115i)10-s − 0.603i·11-s + (0.211 + 0.366i)12-s + 0.960i·13-s + (−0.267 − 0.997i)14-s − 0.189·15-s + (0.499 − 0.866i)16-s − 0.840·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(0.319401\)
Root analytic conductor: \(0.565156\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :1/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.720745 + 0.298542i\)
\(L(\frac12)\) \(\approx\) \(0.720745 + 0.298542i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
5 \( 1 + iT \)
good3 \( 1 + 0.732iT - 3T^{2} \)
7 \( 1 + 2.73T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 - 7.46iT - 19T^{2} \)
23 \( 1 - 4.19T + 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 - 1.46T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 5.46T + 41T^{2} \)
43 \( 1 + 8.73iT - 43T^{2} \)
47 \( 1 - 6.73T + 47T^{2} \)
53 \( 1 - 4.53iT - 53T^{2} \)
59 \( 1 + 0.535iT - 59T^{2} \)
61 \( 1 - 4.92iT - 61T^{2} \)
67 \( 1 - 7.26iT - 67T^{2} \)
71 \( 1 + 1.46T + 71T^{2} \)
73 \( 1 - 0.535T + 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 - 4.73iT - 83T^{2} \)
89 \( 1 + 4.92T + 89T^{2} \)
97 \( 1 - 6.39T + 97T^{2} \)
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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

−16.30356385407447692418013772610, −15.44664233896455035630180779451, −13.88694866347361277026735889824, −13.09060899340453298665285639609, −12.10058734442960941037406833041, −9.877369627672556812380126500789, −8.658063991760145531382247833125, −7.14791678296403584189372620437, −6.03295757928068588483956934399, −4.07608247633227766546385885342, 3.07834236443164268480890653644, 4.78864208392702509396680983103, 6.83400993459043780951836495240, 9.128918287191284636320662147749, 10.11854950467773633428324413242, 11.06649261557972531905603987407, 12.71051298334065032894209785336, 13.30508377236590120326039936482, 14.99138601992189689706858693787, 15.72772912200684433193535867697

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