Properties

Label 2-40-40.29-c5-0-14
Degree $2$
Conductor $40$
Sign $0.984 + 0.175i$
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  + (−5.32 − 1.90i)2-s + 21.6·3-s + (24.7 + 20.2i)4-s + (36.8 − 42.0i)5-s + (−115. − 41.2i)6-s + 236. i·7-s + (−93.4 − 155. i)8-s + 227.·9-s + (−276. + 154. i)10-s − 192. i·11-s + (537. + 439. i)12-s + 975.·13-s + (449. − 1.25e3i)14-s + (798. − 911. i)15-s + (203. + 1.00e3i)16-s − 670. i·17-s + ⋯
L(s)  = 1  + (−0.941 − 0.336i)2-s + 1.39·3-s + (0.774 + 0.632i)4-s + (0.658 − 0.752i)5-s + (−1.31 − 0.467i)6-s + 1.82i·7-s + (−0.516 − 0.856i)8-s + 0.934·9-s + (−0.873 + 0.487i)10-s − 0.480i·11-s + (1.07 + 0.880i)12-s + 1.60·13-s + (0.612 − 1.71i)14-s + (0.916 − 1.04i)15-s + (0.198 + 0.980i)16-s − 0.563i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.984 + 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.984 + 0.175i$
Analytic conductor: \(6.41535\)
Root analytic conductor: \(2.53285\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :5/2),\ 0.984 + 0.175i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.82212 - 0.161435i\)
\(L(\frac12)\) \(\approx\) \(1.82212 - 0.161435i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (5.32 + 1.90i)T \)
5 \( 1 + (-36.8 + 42.0i)T \)
good3 \( 1 - 21.6T + 243T^{2} \)
7 \( 1 - 236. iT - 1.68e4T^{2} \)
11 \( 1 + 192. iT - 1.61e5T^{2} \)
13 \( 1 - 975.T + 3.71e5T^{2} \)
17 \( 1 + 670. iT - 1.41e6T^{2} \)
19 \( 1 + 456. iT - 2.47e6T^{2} \)
23 \( 1 - 2.02e3iT - 6.43e6T^{2} \)
29 \( 1 + 2.78e3iT - 2.05e7T^{2} \)
31 \( 1 + 963.T + 2.86e7T^{2} \)
37 \( 1 + 8.72e3T + 6.93e7T^{2} \)
41 \( 1 + 1.81e3T + 1.15e8T^{2} \)
43 \( 1 - 254.T + 1.47e8T^{2} \)
47 \( 1 - 1.97e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.30e4T + 4.18e8T^{2} \)
59 \( 1 + 2.46e4iT - 7.14e8T^{2} \)
61 \( 1 + 189. iT - 8.44e8T^{2} \)
67 \( 1 + 2.49e4T + 1.35e9T^{2} \)
71 \( 1 + 3.82e4T + 1.80e9T^{2} \)
73 \( 1 - 4.47e4iT - 2.07e9T^{2} \)
79 \( 1 - 6.97e4T + 3.07e9T^{2} \)
83 \( 1 + 4.33e3T + 3.93e9T^{2} \)
89 \( 1 + 5.56e3T + 5.58e9T^{2} \)
97 \( 1 + 9.81e4iT - 8.58e9T^{2} \)
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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

−15.43629928640353066505646848465, −13.82805044971652061657228484482, −12.75597161999559697880517851960, −11.43141523677934018527328749707, −9.499807467673270578790623352376, −8.872530549199879935551826470766, −8.219353043520250821144791808673, −5.94866974512898288363901330257, −3.08464121552711284296559426505, −1.77250721948521934961081912149, 1.60562943543405900870669203094, 3.51752156988372951310452382058, 6.54925853698597363173505274151, 7.62005399253964578010403571182, 8.812579491747654387799728543676, 10.16749257544563221855679117237, 10.79540109516690711874732337597, 13.41517363650773640051406944693, 14.14742862346561858593989323935, 15.02139199389594434650377243382

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