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Properties

Label 2-40-40.37-c2-0-0
Degree $2$
Conductor $40$
Sign $0.572 - 0.819i$
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  + (−1.96 + 0.376i)2-s + (0.977 + 0.977i)3-s + (3.71 − 1.47i)4-s + (−0.801 + 4.93i)5-s + (−2.28 − 1.55i)6-s + (8.39 + 8.39i)7-s + (−6.74 + 4.30i)8-s − 7.08i·9-s + (−0.284 − 9.99i)10-s − 4.01i·11-s + (5.07 + 2.18i)12-s + (−11.0 − 11.0i)13-s + (−19.6 − 13.3i)14-s + (−5.60 + 4.04i)15-s + (11.6 − 10.9i)16-s + (−3.79 − 3.79i)17-s + ⋯
L(s)  = 1  + (−0.982 + 0.188i)2-s + (0.325 + 0.325i)3-s + (0.929 − 0.369i)4-s + (−0.160 + 0.987i)5-s + (−0.381 − 0.258i)6-s + (1.19 + 1.19i)7-s + (−0.842 + 0.538i)8-s − 0.787i·9-s + (−0.0284 − 0.999i)10-s − 0.365i·11-s + (0.423 + 0.182i)12-s + (−0.852 − 0.852i)13-s + (−1.40 − 0.952i)14-s + (−0.373 + 0.269i)15-s + (0.726 − 0.687i)16-s + (−0.223 − 0.223i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $0.572 - 0.819i$
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: \((2,\ 40,\ (\ :1),\ 0.572 - 0.819i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.728087 + 0.379440i\)
\(L(\frac12)\) \(\approx\) \(0.728087 + 0.379440i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.96 - 0.376i)T \)
5 \( 1 + (0.801 - 4.93i)T \)
good3 \( 1 + (-0.977 - 0.977i)T + 9iT^{2} \)
7 \( 1 + (-8.39 - 8.39i)T + 49iT^{2} \)
11 \( 1 + 4.01iT - 121T^{2} \)
13 \( 1 + (11.0 + 11.0i)T + 169iT^{2} \)
17 \( 1 + (3.79 + 3.79i)T + 289iT^{2} \)
19 \( 1 - 15.9T + 361T^{2} \)
23 \( 1 + (-1.86 + 1.86i)T - 529iT^{2} \)
29 \( 1 + 0.468T + 841T^{2} \)
31 \( 1 + 17.3T + 961T^{2} \)
37 \( 1 + (-22.1 + 22.1i)T - 1.36e3iT^{2} \)
41 \( 1 - 37.0T + 1.68e3T^{2} \)
43 \( 1 + (17.1 + 17.1i)T + 1.84e3iT^{2} \)
47 \( 1 + (6.31 + 6.31i)T + 2.20e3iT^{2} \)
53 \( 1 + (39.8 + 39.8i)T + 2.80e3iT^{2} \)
59 \( 1 - 50.6T + 3.48e3T^{2} \)
61 \( 1 - 73.6iT - 3.72e3T^{2} \)
67 \( 1 + (77.6 - 77.6i)T - 4.48e3iT^{2} \)
71 \( 1 + 78.3T + 5.04e3T^{2} \)
73 \( 1 + (46.0 - 46.0i)T - 5.32e3iT^{2} \)
79 \( 1 - 31.6iT - 6.24e3T^{2} \)
83 \( 1 + (-84.9 - 84.9i)T + 6.88e3iT^{2} \)
89 \( 1 + 92.8iT - 7.92e3T^{2} \)
97 \( 1 + (85.3 + 85.3i)T + 9.40e3iT^{2} \)
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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

−15.92487894084579303689501918356, −14.93200620850232596676388902214, −14.57719169541953968553787245566, −12.03908203402306474565663879204, −11.16602204789449274440304601155, −9.815282873834098907717211867657, −8.621694223771485524766838757256, −7.41480924646251503671946136048, −5.69822813448357183650096673658, −2.73203339004127467286042670167, 1.61840469457882146303169509298, 4.64895682750197650604453595481, 7.36263101960122504856868056212, 8.031401428184193773325856747379, 9.428582085973932025583416684951, 10.84660326870164440151512645452, 11.94642182721100631186389956628, 13.35254595850387471962253621893, 14.59378114489539997826701686204, 16.26125780390559840544619392194

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