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Properties

Label 2-40-40.29-c5-0-6
Degree $2$
Conductor $40$
Sign $0.997 + 0.0703i$
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  + (−3.53 − 4.41i)2-s − 21.4·3-s + (−7.01 + 31.2i)4-s + (−48.3 − 27.9i)5-s + (75.7 + 94.6i)6-s + 39.9i·7-s + (162. − 79.4i)8-s + 216.·9-s + (47.3 + 312. i)10-s − 141. i·11-s + (150. − 669. i)12-s + 700.·13-s + (176. − 141. i)14-s + (1.03e3 + 600. i)15-s + (−925. − 437. i)16-s − 960. i·17-s + ⋯
L(s)  = 1  + (−0.624 − 0.780i)2-s − 1.37·3-s + (−0.219 + 0.975i)4-s + (−0.865 − 0.500i)5-s + (0.859 + 1.07i)6-s + 0.307i·7-s + (0.898 − 0.438i)8-s + 0.890·9-s + (0.149 + 0.988i)10-s − 0.352i·11-s + (0.301 − 1.34i)12-s + 1.14·13-s + (0.240 − 0.192i)14-s + (1.19 + 0.688i)15-s + (−0.904 − 0.427i)16-s − 0.805i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0703i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.997 + 0.0703i$
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: \((2,\ 40,\ (\ :5/2),\ 0.997 + 0.0703i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.485772 - 0.0171116i\)
\(L(\frac12)\) \(\approx\) \(0.485772 - 0.0171116i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.53 + 4.41i)T \)
5 \( 1 + (48.3 + 27.9i)T \)
good3 \( 1 + 21.4T + 243T^{2} \)
7 \( 1 - 39.9iT - 1.68e4T^{2} \)
11 \( 1 + 141. iT - 1.61e5T^{2} \)
13 \( 1 - 700.T + 3.71e5T^{2} \)
17 \( 1 + 960. iT - 1.41e6T^{2} \)
19 \( 1 - 2.20e3iT - 2.47e6T^{2} \)
23 \( 1 - 4.49e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.83e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.12e3T + 2.86e7T^{2} \)
37 \( 1 + 8.77e3T + 6.93e7T^{2} \)
41 \( 1 - 1.15e4T + 1.15e8T^{2} \)
43 \( 1 - 1.73e4T + 1.47e8T^{2} \)
47 \( 1 + 1.51e4iT - 2.29e8T^{2} \)
53 \( 1 - 7.11e3T + 4.18e8T^{2} \)
59 \( 1 - 4.17e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.03e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.37e4T + 1.35e9T^{2} \)
71 \( 1 - 881.T + 1.80e9T^{2} \)
73 \( 1 - 7.05e3iT - 2.07e9T^{2} \)
79 \( 1 + 4.77e4T + 3.07e9T^{2} \)
83 \( 1 - 5.16e4T + 3.93e9T^{2} \)
89 \( 1 - 1.22e5T + 5.58e9T^{2} \)
97 \( 1 - 1.55e5iT - 8.58e9T^{2} \)
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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

−15.80494333904676393480120781385, −13.44723917423926206020299104639, −12.08074888681862172645602502005, −11.62321855061938074244382311624, −10.58912170908092900929583787932, −8.987843042213996313359637702356, −7.60216726722799454064545502658, −5.64906655812530673982430683540, −3.86494953716741814962430522231, −0.954782224919975861455689005886, 0.57076565437535299518996939084, 4.54622146488406913421186200939, 6.18384662050465822456551439982, 7.11817649235771003931360503563, 8.666173904774244904745795141891, 10.68490074960147808249653543596, 10.99436690753870478346718883453, 12.59656848762156798352633614594, 14.34917559594029590461528725319, 15.61533747059390916582101906798

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