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Properties

Label 2-40-40.27-c1-0-2
Degree $2$
Conductor $40$
Sign $0.987 + 0.156i$
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  + (1.26 − 0.642i)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 + (0.442 − 2.79i)8-s − 2.23i·9-s + (−3.15 − 0.260i)10-s + 1.23·11-s + (0.715 + 4.52i)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  + (0.891 − 0.453i)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 + (0.156 − 0.987i)8-s − 0.745i·9-s + (−0.996 − 0.0822i)10-s + 0.372·11-s + (0.206 + 1.30i)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

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

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.987 + 0.156i$
Analytic conductor: \(0.319401\)
Root analytic conductor: \(0.565156\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :1/2),\ 0.987 + 0.156i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.861711 - 0.0678181i\)
\(L(\frac12)\) \(\approx\) \(0.861711 - 0.0678181i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.26 + 0.642i)T \)
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

−15.94819785570695514273092154849, −15.37062844682829259638441035257, −13.76171232546356373957128449735, −12.26755018430091146729133468359, −11.57321441044225953845181801254, −10.53040438601228110677526502787, −9.041697321330221590405612490417, −6.52421587298763225381509625730, −5.04691926888097468421191304043, −3.89312045284480504322586550705, 3.73976913195957760440073579450, 5.89554168775237004951217578524, 6.88986761452075623932627357931, 7.987190908828908892803373876433, 10.78892239218923390152901247382, 11.77497760147220932639089438707, 12.68801909886942031682014282319, 13.73018927244390833543311665805, 15.14797011358104720587035570622, 16.15117989859528759182721471688

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