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Properties

Label 2-40-40.13-c4-0-15
Degree $2$
Conductor $40$
Sign $0.964 - 0.264i$
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  + (3.32 + 2.22i)2-s + (9.32 − 9.32i)3-s + (6.10 + 14.7i)4-s + (−13.0 + 21.2i)5-s + (51.7 − 10.2i)6-s + (43.0 − 43.0i)7-s + (−12.5 + 62.7i)8-s − 93.0i·9-s + (−90.8 + 41.6i)10-s − 128. i·11-s + (194. + 81.0i)12-s + (−193. + 193. i)13-s + (238. − 47.3i)14-s + (76.5 + 320. i)15-s + (−181. + 180. i)16-s + (41.5 − 41.5i)17-s + ⋯
L(s)  = 1  + (0.831 + 0.556i)2-s + (1.03 − 1.03i)3-s + (0.381 + 0.924i)4-s + (−0.523 + 0.851i)5-s + (1.43 − 0.285i)6-s + (0.877 − 0.877i)7-s + (−0.196 + 0.980i)8-s − 1.14i·9-s + (−0.908 + 0.416i)10-s − 1.06i·11-s + (1.35 + 0.562i)12-s + (−1.14 + 1.14i)13-s + (1.21 − 0.241i)14-s + (0.340 + 1.42i)15-s + (−0.708 + 0.705i)16-s + (0.143 − 0.143i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.964 - 0.264i$
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: \((2,\ 40,\ (\ :2),\ 0.964 - 0.264i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.78272 + 0.375156i\)
\(L(\frac12)\) \(\approx\) \(2.78272 + 0.375156i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.32 - 2.22i)T \)
5 \( 1 + (13.0 - 21.2i)T \)
good3 \( 1 + (-9.32 + 9.32i)T - 81iT^{2} \)
7 \( 1 + (-43.0 + 43.0i)T - 2.40e3iT^{2} \)
11 \( 1 + 128. iT - 1.46e4T^{2} \)
13 \( 1 + (193. - 193. i)T - 2.85e4iT^{2} \)
17 \( 1 + (-41.5 + 41.5i)T - 8.35e4iT^{2} \)
19 \( 1 + 480.T + 1.30e5T^{2} \)
23 \( 1 + (89.7 + 89.7i)T + 2.79e5iT^{2} \)
29 \( 1 - 449.T + 7.07e5T^{2} \)
31 \( 1 - 264.T + 9.23e5T^{2} \)
37 \( 1 + (-46.3 - 46.3i)T + 1.87e6iT^{2} \)
41 \( 1 - 2.33e3T + 2.82e6T^{2} \)
43 \( 1 + (38.4 - 38.4i)T - 3.41e6iT^{2} \)
47 \( 1 + (-781. + 781. i)T - 4.87e6iT^{2} \)
53 \( 1 + (114. - 114. i)T - 7.89e6iT^{2} \)
59 \( 1 - 5.36e3T + 1.21e7T^{2} \)
61 \( 1 - 2.43e3iT - 1.38e7T^{2} \)
67 \( 1 + (2.17e3 + 2.17e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 3.68e3T + 2.54e7T^{2} \)
73 \( 1 + (-3.99e3 - 3.99e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 473. iT - 3.89e7T^{2} \)
83 \( 1 + (-5.00e3 + 5.00e3i)T - 4.74e7iT^{2} \)
89 \( 1 - 1.13e4iT - 6.27e7T^{2} \)
97 \( 1 + (1.64e3 - 1.64e3i)T - 8.85e7iT^{2} \)
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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

−14.79021092401261072695425930171, −14.29504085289956697877126057118, −13.57609112373834329485842769282, −12.15481861352387864365336566225, −10.98802301121194679470497905477, −8.445819280634447183457571385871, −7.53854655442891921634620484245, −6.66093070020549773678129958277, −4.16948799383425264543495437689, −2.48706681068675341325068339642, 2.41268719452811805101714914414, 4.25740249272585607305321455468, 5.15059761767192361986731825739, 8.001652201161959813240794045218, 9.273996783850958764285988038112, 10.40835945655302284160065344362, 12.01321757389687896226549150888, 12.84426120423061577196534044128, 14.62129158229879162809249653268, 15.02153254559828583974642684607

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