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Properties

Label 2-40-40.13-c4-0-21
Degree $2$
Conductor $40$
Sign $-0.944 - 0.329i$
Analytic cond. $4.13479$
Root an. cond. $2.03342$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.35 − 3.23i)2-s + (−9.60 + 9.60i)3-s + (−4.89 − 15.2i)4-s + (−12.7 − 21.4i)5-s + (8.41 + 53.6i)6-s + (−60.7 + 60.7i)7-s + (−60.7 − 20.0i)8-s − 103. i·9-s + (−99.5 − 9.24i)10-s − 85.0i·11-s + (193. + 99.2i)12-s + (−41.8 + 41.8i)13-s + (53.2 + 339. i)14-s + (329. + 83.3i)15-s + (−208. + 149. i)16-s + (181. − 181. i)17-s + ⋯
L(s)  = 1  + (0.589 − 0.808i)2-s + (−1.06 + 1.06i)3-s + (−0.305 − 0.952i)4-s + (−0.511 − 0.859i)5-s + (0.233 + 1.49i)6-s + (−1.24 + 1.24i)7-s + (−0.949 − 0.313i)8-s − 1.27i·9-s + (−0.995 − 0.0924i)10-s − 0.703i·11-s + (1.34 + 0.689i)12-s + (−0.247 + 0.247i)13-s + (0.271 + 1.73i)14-s + (1.46 + 0.370i)15-s + (−0.812 + 0.582i)16-s + (0.629 − 0.629i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.944 - 0.329i$
Analytic conductor: \(4.13479\)
Root analytic conductor: \(2.03342\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :2),\ -0.944 - 0.329i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0200072 + 0.118237i\)
\(L(\frac12)\) \(\approx\) \(0.0200072 + 0.118237i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-2.35 + 3.23i)T \)
5 \( 1 + (12.7 + 21.4i)T \)
good3 \( 1 + (9.60 - 9.60i)T - 81iT^{2} \)
7 \( 1 + (60.7 - 60.7i)T - 2.40e3iT^{2} \)
11 \( 1 + 85.0iT - 1.46e4T^{2} \)
13 \( 1 + (41.8 - 41.8i)T - 2.85e4iT^{2} \)
17 \( 1 + (-181. + 181. i)T - 8.35e4iT^{2} \)
19 \( 1 - 260.T + 1.30e5T^{2} \)
23 \( 1 + (180. + 180. i)T + 2.79e5iT^{2} \)
29 \( 1 + 1.11e3T + 7.07e5T^{2} \)
31 \( 1 + 571.T + 9.23e5T^{2} \)
37 \( 1 + (-273. - 273. i)T + 1.87e6iT^{2} \)
41 \( 1 + 503.T + 2.82e6T^{2} \)
43 \( 1 + (2.05e3 - 2.05e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-66.4 + 66.4i)T - 4.87e6iT^{2} \)
53 \( 1 + (-518. + 518. i)T - 7.89e6iT^{2} \)
59 \( 1 - 5.24e3T + 1.21e7T^{2} \)
61 \( 1 + 2.27e3iT - 1.38e7T^{2} \)
67 \( 1 + (4.86e3 + 4.86e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 7.97e3T + 2.54e7T^{2} \)
73 \( 1 + (-293. - 293. i)T + 2.83e7iT^{2} \)
79 \( 1 + 4.82e3iT - 3.89e7T^{2} \)
83 \( 1 + (748. - 748. i)T - 4.74e7iT^{2} \)
89 \( 1 - 6.61e3iT - 6.27e7T^{2} \)
97 \( 1 + (-7.80e3 + 7.80e3i)T - 8.85e7iT^{2} \)
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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

−15.00227502240806929401223733501, −13.14216140467558901024510608943, −12.04599048282674610426443606955, −11.45895805402192486316551589197, −9.884333494080606990125857869308, −9.114475635150264922617332883733, −5.93334637225387818541297815487, −5.07773734468923874867092378525, −3.46556454780961651405881673835, −0.07247214963876999379664200159, 3.67513532838748147059130125847, 5.84772583623370972802895862323, 7.07983177281731301571812217515, 7.42639108758966749515516015086, 10.14457019398322601493077901572, 11.65449688366583057444575854182, 12.68957122098180778148804704308, 13.51566080868569444006051660818, 14.86398453080650631058220347626, 16.22609900670074821800946050486

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