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Properties

Label 2-40-40.37-c2-0-1
Degree $2$
Conductor $40$
Sign $-0.0242 - 0.999i$
Analytic cond. $1.08992$
Root an. cond. $1.04399$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.665 + 1.88i)2-s + (3.60 + 3.60i)3-s + (−3.11 − 2.51i)4-s + (−2.34 − 4.41i)5-s + (−9.20 + 4.40i)6-s + (1.47 + 1.47i)7-s + (6.80 − 4.20i)8-s + 17.0i·9-s + (9.88 − 1.48i)10-s − 11.3i·11-s + (−2.18 − 20.2i)12-s + (3.17 + 3.17i)13-s + (−3.77 + 1.80i)14-s + (7.46 − 24.3i)15-s + (3.39 + 15.6i)16-s + (−9.94 − 9.94i)17-s + ⋯
L(s)  = 1  + (−0.332 + 0.943i)2-s + (1.20 + 1.20i)3-s + (−0.778 − 0.627i)4-s + (−0.469 − 0.883i)5-s + (−1.53 + 0.733i)6-s + (0.211 + 0.211i)7-s + (0.850 − 0.525i)8-s + 1.89i·9-s + (0.988 − 0.148i)10-s − 1.02i·11-s + (−0.181 − 1.69i)12-s + (0.244 + 0.244i)13-s + (−0.269 + 0.128i)14-s + (0.497 − 1.62i)15-s + (0.212 + 0.977i)16-s + (−0.585 − 0.585i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0242 - 0.999i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0242 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.0242 - 0.999i$
Analytic conductor: \(1.08992\)
Root analytic conductor: \(1.04399\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{40} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 40,\ (\ :1),\ -0.0242 - 0.999i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.786780 + 0.806131i\)
\(L(\frac12)\) \(\approx\) \(0.786780 + 0.806131i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.665 - 1.88i)T \)
5 \( 1 + (2.34 + 4.41i)T \)
good3 \( 1 + (-3.60 - 3.60i)T + 9iT^{2} \)
7 \( 1 + (-1.47 - 1.47i)T + 49iT^{2} \)
11 \( 1 + 11.3iT - 121T^{2} \)
13 \( 1 + (-3.17 - 3.17i)T + 169iT^{2} \)
17 \( 1 + (9.94 + 9.94i)T + 289iT^{2} \)
19 \( 1 - 11.2T + 361T^{2} \)
23 \( 1 + (-1.67 + 1.67i)T - 529iT^{2} \)
29 \( 1 + 41.1T + 841T^{2} \)
31 \( 1 + 29.2T + 961T^{2} \)
37 \( 1 + (-8.60 + 8.60i)T - 1.36e3iT^{2} \)
41 \( 1 + 19.6T + 1.68e3T^{2} \)
43 \( 1 + (-25.1 - 25.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (-41.7 - 41.7i)T + 2.20e3iT^{2} \)
53 \( 1 + (2.16 + 2.16i)T + 2.80e3iT^{2} \)
59 \( 1 + 38.9T + 3.48e3T^{2} \)
61 \( 1 - 87.5iT - 3.72e3T^{2} \)
67 \( 1 + (-31.1 + 31.1i)T - 4.48e3iT^{2} \)
71 \( 1 - 134.T + 5.04e3T^{2} \)
73 \( 1 + (26.1 - 26.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 23.1iT - 6.24e3T^{2} \)
83 \( 1 + (-68.3 - 68.3i)T + 6.88e3iT^{2} \)
89 \( 1 + 75.2iT - 7.92e3T^{2} \)
97 \( 1 + (-57.5 - 57.5i)T + 9.40e3iT^{2} \)
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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

−16.07777353959742085805095212288, −15.32592692504584982730084812473, −14.24741543844654770597697283314, −13.29553320211968589411967854502, −11.06176399351555750560206940570, −9.389167021615519454150414896763, −8.834995416467595895501065831178, −7.77381618285633683633626704079, −5.29856854435201720432563050925, −3.92726453302330650844516521120, 2.07995624338543992283468207048, 3.64595090523606552170010035148, 7.15236491757152050546986656613, 7.966376461843252639970224636735, 9.334764786167300139979279527993, 10.86454961618627660147231789022, 12.20440563218956398657937915630, 13.17982290420087956360123876789, 14.20594779635437694514919023627, 15.18962655491925691943892456207

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