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Properties

Label 2-40-8.5-c5-0-11
Degree $2$
Conductor $40$
Sign $0.999 - 0.0262i$
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  + (0.0494 + 5.65i)2-s − 10.7i·3-s + (−31.9 + 0.559i)4-s − 25i·5-s + (60.7 − 0.531i)6-s + 198.·7-s + (−4.74 − 180. i)8-s + 127.·9-s + (141. − 1.23i)10-s − 85.9i·11-s + (6.01 + 343. i)12-s − 407. i·13-s + (9.83 + 1.12e3i)14-s − 268.·15-s + (1.02e3 − 35.8i)16-s + 1.20e3·17-s + ⋯
L(s)  = 1  + (0.00874 + 0.999i)2-s − 0.689i·3-s + (−0.999 + 0.0174i)4-s − 0.447i·5-s + (0.689 − 0.00602i)6-s + 1.53·7-s + (−0.0262 − 0.999i)8-s + 0.524·9-s + (0.447 − 0.00391i)10-s − 0.214i·11-s + (0.0120 + 0.689i)12-s − 0.668i·13-s + (0.0134 + 1.53i)14-s − 0.308·15-s + (0.999 − 0.0349i)16-s + 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0262i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0262i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.999 - 0.0262i$
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: \((2,\ 40,\ (\ :5/2),\ 0.999 - 0.0262i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.68967 + 0.0221702i\)
\(L(\frac12)\) \(\approx\) \(1.68967 + 0.0221702i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0494 - 5.65i)T \)
5 \( 1 + 25iT \)
good3 \( 1 + 10.7iT - 243T^{2} \)
7 \( 1 - 198.T + 1.68e4T^{2} \)
11 \( 1 + 85.9iT - 1.61e5T^{2} \)
13 \( 1 + 407. iT - 3.71e5T^{2} \)
17 \( 1 - 1.20e3T + 1.41e6T^{2} \)
19 \( 1 - 206. iT - 2.47e6T^{2} \)
23 \( 1 + 2.59e3T + 6.43e6T^{2} \)
29 \( 1 + 6.19e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.86e3T + 2.86e7T^{2} \)
37 \( 1 - 1.47e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.80e4T + 1.15e8T^{2} \)
43 \( 1 - 9.26e3iT - 1.47e8T^{2} \)
47 \( 1 + 2.43e4T + 2.29e8T^{2} \)
53 \( 1 - 1.27e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.07e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.13e4iT - 8.44e8T^{2} \)
67 \( 1 - 6.26e4iT - 1.35e9T^{2} \)
71 \( 1 + 6.12e4T + 1.80e9T^{2} \)
73 \( 1 - 2.32e4T + 2.07e9T^{2} \)
79 \( 1 - 2.91e4T + 3.07e9T^{2} \)
83 \( 1 + 4.80e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.01e4T + 5.58e9T^{2} \)
97 \( 1 + 1.13e5T + 8.58e9T^{2} \)
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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

−15.09076834302304086982255081775, −14.08762265228919770765945838604, −13.03291455163885037720912771417, −11.84293611097104954255846776616, −9.969692525099070805420964102474, −8.189689846030477180178868862217, −7.68522175019223183184109891378, −5.89017948863040940061067559255, −4.50160930087904594174855467603, −1.18163940505072871177166965518, 1.75341189498998976147199956875, 3.90744714601063730703799695672, 5.12268857001657364394650960783, 7.72665860462029115285702781507, 9.244164907051766853331155633014, 10.45308157249826425327899799114, 11.28941013375291602322210236281, 12.47562789300324280504342583796, 14.19022828430160957722805490122, 14.70147901918569284439048223952

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