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Properties

Label 2-40-40.37-c2-0-4
Degree $2$
Conductor $40$
Sign $0.901 - 0.432i$
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  + (1.84 + 0.764i)2-s + (0.130 + 0.130i)3-s + (2.83 + 2.82i)4-s + (−4.38 − 2.39i)5-s + (0.141 + 0.341i)6-s + (−1.59 − 1.59i)7-s + (3.07 + 7.38i)8-s − 8.96i·9-s + (−6.27 − 7.78i)10-s + 11.9i·11-s + (0.000715 + 0.739i)12-s + (−9.59 − 9.59i)13-s + (−1.73 − 4.17i)14-s + (−0.260 − 0.887i)15-s + (0.0309 + 15.9i)16-s + (0.857 + 0.857i)17-s + ⋯
L(s)  = 1  + (0.924 + 0.382i)2-s + (0.0435 + 0.0435i)3-s + (0.707 + 0.706i)4-s + (−0.877 − 0.479i)5-s + (0.0236 + 0.0569i)6-s + (−0.228 − 0.228i)7-s + (0.384 + 0.923i)8-s − 0.996i·9-s + (−0.627 − 0.778i)10-s + 1.08i·11-s + (5.96e−5 + 0.0616i)12-s + (−0.738 − 0.738i)13-s + (−0.123 − 0.298i)14-s + (−0.0173 − 0.0591i)15-s + (0.00193 + 0.999i)16-s + (0.0504 + 0.0504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.901 - 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.901 - 0.432i$
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: \((2,\ 40,\ (\ :1),\ 0.901 - 0.432i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.48238 + 0.337311i\)
\(L(\frac12)\) \(\approx\) \(1.48238 + 0.337311i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.84 - 0.764i)T \)
5 \( 1 + (4.38 + 2.39i)T \)
good3 \( 1 + (-0.130 - 0.130i)T + 9iT^{2} \)
7 \( 1 + (1.59 + 1.59i)T + 49iT^{2} \)
11 \( 1 - 11.9iT - 121T^{2} \)
13 \( 1 + (9.59 + 9.59i)T + 169iT^{2} \)
17 \( 1 + (-0.857 - 0.857i)T + 289iT^{2} \)
19 \( 1 - 20.5T + 361T^{2} \)
23 \( 1 + (22.1 - 22.1i)T - 529iT^{2} \)
29 \( 1 - 27.3T + 841T^{2} \)
31 \( 1 - 40.0T + 961T^{2} \)
37 \( 1 + (-1.57 + 1.57i)T - 1.36e3iT^{2} \)
41 \( 1 + 37.5T + 1.68e3T^{2} \)
43 \( 1 + (49.2 + 49.2i)T + 1.84e3iT^{2} \)
47 \( 1 + (34.0 + 34.0i)T + 2.20e3iT^{2} \)
53 \( 1 + (-28.8 - 28.8i)T + 2.80e3iT^{2} \)
59 \( 1 + 92.7T + 3.48e3T^{2} \)
61 \( 1 + 4.82iT - 3.72e3T^{2} \)
67 \( 1 + (-54.6 + 54.6i)T - 4.48e3iT^{2} \)
71 \( 1 - 59.2T + 5.04e3T^{2} \)
73 \( 1 + (-34.1 + 34.1i)T - 5.32e3iT^{2} \)
79 \( 1 - 96.2iT - 6.24e3T^{2} \)
83 \( 1 + (-63.6 - 63.6i)T + 6.88e3iT^{2} \)
89 \( 1 - 3.68iT - 7.92e3T^{2} \)
97 \( 1 + (-46.0 - 46.0i)T + 9.40e3iT^{2} \)
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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

−15.55736036336381116835310960207, −15.17090344642307432364496479290, −13.70427875812231337381455650983, −12.29752307374262690303120163233, −11.91424889744147014555940681592, −9.896169047837992760796656273387, −8.044517119096137809351907340033, −6.89372702050246015240894428474, −5.01729033259692478271841956438, −3.54317213834937212926206270373, 2.93044253671158623833538021946, 4.70356988737755394919511362160, 6.49323329994369630047032786499, 8.009665073603227448682458442476, 10.12505294925913118217105055104, 11.34366357505020612632215986682, 12.14647353386081955921892293157, 13.67980249329372631275721287961, 14.42125581003900451658085932433, 15.82365832760389097764483772320

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