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Properties

Label 2-40-8.5-c1-0-2
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  + (−1.36 + 0.366i)2-s − 2.73i·3-s + (1.73 − i)4-s + i·5-s + (1 + 3.73i)6-s + 0.732·7-s + (−1.99 + 2i)8-s − 4.46·9-s + (−0.366 − 1.36i)10-s + 2i·11-s + (−2.73 − 4.73i)12-s + 3.46i·13-s + (−1 + 0.267i)14-s + 2.73·15-s + (1.99 − 3.46i)16-s + 3.46·17-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)2-s − 1.57i·3-s + (0.866 − 0.5i)4-s + 0.447i·5-s + (0.408 + 1.52i)6-s + 0.276·7-s + (−0.707 + 0.707i)8-s − 1.48·9-s + (−0.115 − 0.431i)10-s + 0.603i·11-s + (−0.788 − 1.36i)12-s + 0.960i·13-s + (−0.267 + 0.0716i)14-s + 0.705·15-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

\(L(1)\) \(\approx\) \(0.502887 - 0.208302i\)
\(L(\frac12)\) \(\approx\) \(0.502887 - 0.208302i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.366i)T \)
5 \( 1 - iT \)
good3 \( 1 + 2.73iT - 3T^{2} \)
7 \( 1 - 0.732T + 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 0.535iT - 19T^{2} \)
23 \( 1 + 6.19T + 23T^{2} \)
29 \( 1 + 6.92iT - 29T^{2} \)
31 \( 1 + 5.46T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 1.46T + 41T^{2} \)
43 \( 1 - 5.26iT - 43T^{2} \)
47 \( 1 - 3.26T + 47T^{2} \)
53 \( 1 + 11.4iT - 53T^{2} \)
59 \( 1 - 7.46iT - 59T^{2} \)
61 \( 1 - 8.92iT - 61T^{2} \)
67 \( 1 + 10.7iT - 67T^{2} \)
71 \( 1 - 5.46T + 71T^{2} \)
73 \( 1 - 7.46T + 73T^{2} \)
79 \( 1 + 1.07T + 79T^{2} \)
83 \( 1 + 1.26iT - 83T^{2} \)
89 \( 1 - 8.92T + 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−16.42173613282568116020136566766, −14.83355699707662702739601384107, −13.85506122061115528653571763758, −12.26929008658477693804126621085, −11.40092958554935680105153880476, −9.759763384450441618832338961039, −8.116337803120132320517287048108, −7.22925099472382298278932711966, −6.13409915347686679763897898118, −1.98800828619583758752930483423, 3.55743468534700640706687531154, 5.53425094987208521971798192883, 8.003458413310215123129012497393, 9.130388608674758292080000760602, 10.20076701750556844782565606349, 11.03942142649809864668539328989, 12.42945650528245039176314554362, 14.44816501107516912574810046249, 15.68749848078677844496816366876, 16.33247980106977342435391363008

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