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Properties

Label 2-40-40.13-c4-0-16
Degree $2$
Conductor $40$
Sign $0.880 + 0.474i$
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  + (3.99 + 0.133i)2-s + (4.11 − 4.11i)3-s + (15.9 + 1.06i)4-s + (−3.95 − 24.6i)5-s + (16.9 − 15.8i)6-s + (−12.8 + 12.8i)7-s + (63.6 + 6.41i)8-s + 47.1i·9-s + (−12.5 − 99.2i)10-s + 28.7i·11-s + (70.0 − 61.2i)12-s + (21.7 − 21.7i)13-s + (−53.1 + 49.6i)14-s + (−117. − 85.2i)15-s + (253. + 34.1i)16-s + (−308. + 308. i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.0334i)2-s + (0.457 − 0.457i)3-s + (0.997 + 0.0668i)4-s + (−0.158 − 0.987i)5-s + (0.472 − 0.441i)6-s + (−0.262 + 0.262i)7-s + (0.994 + 0.100i)8-s + 0.582i·9-s + (−0.125 − 0.992i)10-s + 0.237i·11-s + (0.486 − 0.425i)12-s + (0.128 − 0.128i)13-s + (−0.270 + 0.253i)14-s + (−0.523 − 0.379i)15-s + (0.991 + 0.133i)16-s + (−1.06 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.880 + 0.474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.880 + 0.474i$
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: \((2,\ 40,\ (\ :2),\ 0.880 + 0.474i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.69068 - 0.679774i\)
\(L(\frac12)\) \(\approx\) \(2.69068 - 0.679774i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.99 - 0.133i)T \)
5 \( 1 + (3.95 + 24.6i)T \)
good3 \( 1 + (-4.11 + 4.11i)T - 81iT^{2} \)
7 \( 1 + (12.8 - 12.8i)T - 2.40e3iT^{2} \)
11 \( 1 - 28.7iT - 1.46e4T^{2} \)
13 \( 1 + (-21.7 + 21.7i)T - 2.85e4iT^{2} \)
17 \( 1 + (308. - 308. i)T - 8.35e4iT^{2} \)
19 \( 1 - 85.6T + 1.30e5T^{2} \)
23 \( 1 + (329. + 329. i)T + 2.79e5iT^{2} \)
29 \( 1 + 1.28e3T + 7.07e5T^{2} \)
31 \( 1 - 856.T + 9.23e5T^{2} \)
37 \( 1 + (981. + 981. i)T + 1.87e6iT^{2} \)
41 \( 1 - 1.65e3T + 2.82e6T^{2} \)
43 \( 1 + (-2.04e3 + 2.04e3i)T - 3.41e6iT^{2} \)
47 \( 1 + (-2.90e3 + 2.90e3i)T - 4.87e6iT^{2} \)
53 \( 1 + (-648. + 648. i)T - 7.89e6iT^{2} \)
59 \( 1 + 3.54e3T + 1.21e7T^{2} \)
61 \( 1 - 1.60e3iT - 1.38e7T^{2} \)
67 \( 1 + (1.11e3 + 1.11e3i)T + 2.01e7iT^{2} \)
71 \( 1 + 6.64e3T + 2.54e7T^{2} \)
73 \( 1 + (2.15e3 + 2.15e3i)T + 2.83e7iT^{2} \)
79 \( 1 - 9.29e3iT - 3.89e7T^{2} \)
83 \( 1 + (-5.04e3 + 5.04e3i)T - 4.74e7iT^{2} \)
89 \( 1 + 9.90e3iT - 6.27e7T^{2} \)
97 \( 1 + (4.54e3 - 4.54e3i)T - 8.85e7iT^{2} \)
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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

−15.24767034104089314711486636170, −13.86567154578368013039997612974, −13.00200039090651563310788392606, −12.23080348891780591955114227218, −10.71103688548142758468421089241, −8.749356760649084197993707597527, −7.48237313725560091098403032943, −5.76526099729879039108408876870, −4.21272242808076690446289692409, −2.07107586376697846916861619790, 2.87192193142850629162392598941, 4.11625347308948516537288778501, 6.18871111450130174824636856526, 7.41768537482511009789200584634, 9.500943705353422588924618561600, 10.85823793704889086026572141567, 11.87735379554280290440842508796, 13.48504429239889527480069371148, 14.30287953532905047388613397020, 15.34340824842968098509544497939

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