Properties

Label 2-40-8.5-c5-0-15
Degree $2$
Conductor $40$
Sign $-0.993 + 0.116i$
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  + (−4.78 + 3.01i)2-s − 25.4i·3-s + (13.7 − 28.8i)4-s − 25i·5-s + (76.7 + 121. i)6-s − 56.4·7-s + (21.0 + 179. i)8-s − 403.·9-s + (75.4 + 119. i)10-s + 261. i·11-s + (−734. − 350. i)12-s − 720. i·13-s + (270. − 170. i)14-s − 635.·15-s + (−643. − 796. i)16-s − 1.87e3·17-s + ⋯
L(s)  = 1  + (−0.845 + 0.533i)2-s − 1.63i·3-s + (0.431 − 0.902i)4-s − 0.447i·5-s + (0.870 + 1.38i)6-s − 0.435·7-s + (0.116 + 0.993i)8-s − 1.66·9-s + (0.238 + 0.378i)10-s + 0.650i·11-s + (−1.47 − 0.703i)12-s − 1.18i·13-s + (0.368 − 0.232i)14-s − 0.729·15-s + (−0.628 − 0.778i)16-s − 1.57·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.993 + 0.116i$
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: \((2,\ 40,\ (\ :5/2),\ -0.993 + 0.116i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.0318210 - 0.545135i\)
\(L(\frac12)\) \(\approx\) \(0.0318210 - 0.545135i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.78 - 3.01i)T \)
5 \( 1 + 25iT \)
good3 \( 1 + 25.4iT - 243T^{2} \)
7 \( 1 + 56.4T + 1.68e4T^{2} \)
11 \( 1 - 261. iT - 1.61e5T^{2} \)
13 \( 1 + 720. iT - 3.71e5T^{2} \)
17 \( 1 + 1.87e3T + 1.41e6T^{2} \)
19 \( 1 - 1.99e3iT - 2.47e6T^{2} \)
23 \( 1 - 2.57e3T + 6.43e6T^{2} \)
29 \( 1 + 1.70e3iT - 2.05e7T^{2} \)
31 \( 1 + 7.73e3T + 2.86e7T^{2} \)
37 \( 1 + 1.22e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.49e4T + 1.15e8T^{2} \)
43 \( 1 + 1.81e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.14e3T + 2.29e8T^{2} \)
53 \( 1 + 1.60e3iT - 4.18e8T^{2} \)
59 \( 1 + 2.68e3iT - 7.14e8T^{2} \)
61 \( 1 + 4.45e4iT - 8.44e8T^{2} \)
67 \( 1 - 1.24e4iT - 1.35e9T^{2} \)
71 \( 1 - 8.18e3T + 1.80e9T^{2} \)
73 \( 1 + 4.10e4T + 2.07e9T^{2} \)
79 \( 1 - 4.63e4T + 3.07e9T^{2} \)
83 \( 1 + 6.16e4iT - 3.93e9T^{2} \)
89 \( 1 - 5.32e4T + 5.58e9T^{2} \)
97 \( 1 + 3.92e4T + 8.58e9T^{2} \)
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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

−14.63943489922002867465590135111, −13.20677671278805413310611686322, −12.42762060537959105946118417049, −10.85882712534808644129822373702, −9.168248492238149391517204494804, −7.929472615518512249182135200768, −6.97398241242138959002025300517, −5.71396772483353115874274391965, −2.00629340232989276197903019190, −0.38345842016112602331330427621, 2.93664359031754054838196958755, 4.38021881935717754243387996048, 6.73419710463704987266415017807, 8.899016883925746138676638612062, 9.451981900246634135585584144016, 10.91017559716701977948182259737, 11.28257189450929486535582814470, 13.30873577810901352629493080512, 14.94448089229218011456676125215, 15.98169956113949797730976656021

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