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Properties

Label 2-40-8.5-c5-0-14
Degree $2$
Conductor $40$
Sign $0.165 + 0.986i$
Analytic cond. $6.41535$
Root an. cond. $2.53285$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.55 + 5.04i)2-s + 11.5i·3-s + (−18.9 − 25.7i)4-s − 25i·5-s + (−58.5 − 29.5i)6-s − 231.·7-s + (178. − 30.0i)8-s + 108.·9-s + (126. + 63.8i)10-s − 559. i·11-s + (298. − 219. i)12-s + 107. i·13-s + (590. − 1.16e3i)14-s + 289.·15-s + (−303. + 977. i)16-s − 441.·17-s + ⋯
L(s)  = 1  + (−0.451 + 0.892i)2-s + 0.743i·3-s + (−0.592 − 0.805i)4-s − 0.447i·5-s + (−0.663 − 0.335i)6-s − 1.78·7-s + (0.986 − 0.165i)8-s + 0.446·9-s + (0.399 + 0.201i)10-s − 1.39i·11-s + (0.598 − 0.440i)12-s + 0.177i·13-s + (0.805 − 1.59i)14-s + 0.332·15-s + (−0.296 + 0.954i)16-s − 0.370·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.165 + 0.986i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ 0.165 + 0.986i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.227010 - 0.192014i\)
\(L(\frac12)\) \(\approx\) \(0.227010 - 0.192014i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.55 - 5.04i)T \)
5 \( 1 + 25iT \)
good3 \( 1 - 11.5iT - 243T^{2} \)
7 \( 1 + 231.T + 1.68e4T^{2} \)
11 \( 1 + 559. iT - 1.61e5T^{2} \)
13 \( 1 - 107. iT - 3.71e5T^{2} \)
17 \( 1 + 441.T + 1.41e6T^{2} \)
19 \( 1 + 1.87e3iT - 2.47e6T^{2} \)
23 \( 1 + 3.83e3T + 6.43e6T^{2} \)
29 \( 1 - 3.36e3iT - 2.05e7T^{2} \)
31 \( 1 + 7.95e3T + 2.86e7T^{2} \)
37 \( 1 + 1.06e4iT - 6.93e7T^{2} \)
41 \( 1 + 9.96e3T + 1.15e8T^{2} \)
43 \( 1 - 925. iT - 1.47e8T^{2} \)
47 \( 1 - 8.06e3T + 2.29e8T^{2} \)
53 \( 1 + 7.95e3iT - 4.18e8T^{2} \)
59 \( 1 - 1.68e4iT - 7.14e8T^{2} \)
61 \( 1 - 1.12e4iT - 8.44e8T^{2} \)
67 \( 1 - 3.36e4iT - 1.35e9T^{2} \)
71 \( 1 + 8.86e3T + 1.80e9T^{2} \)
73 \( 1 - 5.55e4T + 2.07e9T^{2} \)
79 \( 1 - 6.94e4T + 3.07e9T^{2} \)
83 \( 1 + 1.02e4iT - 3.93e9T^{2} \)
89 \( 1 + 9.24e4T + 5.58e9T^{2} \)
97 \( 1 + 8.86e4T + 8.58e9T^{2} \)
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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

−15.41309796527444405828218430870, −13.79967341562374348552199613935, −12.86196739187090882913390677962, −10.74050044118723786850344830684, −9.582688833605266821696678629459, −8.831934408742884364072461900602, −6.93463274781845467705735517403, −5.65476591215631903428764529245, −3.84703575743602622521395923784, −0.18641905283586113437691629914, 2.05377301906553763848797910005, 3.78946797692754930015173511583, 6.56165790890273842000651158755, 7.71028004550308703075031587290, 9.656552313948985726200364458226, 10.21018169467133402961959511200, 12.15222254885967724941569737699, 12.69940215045393511165502009351, 13.71569033849758422555001914947, 15.53922315848890428059389757913

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