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Properties

Label 2-40-40.37-c2-0-7
Degree $2$
Conductor $40$
Sign $-0.174 + 0.984i$
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  + (0.376 − 1.96i)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 + (−4.30 + 6.74i)8-s − 7.08i·9-s + (−9.39 − 3.43i)10-s + 4.01i·11-s + (2.18 + 5.07i)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  + (0.188 − 0.982i)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 + (−0.538 + 0.842i)8-s − 0.787i·9-s + (−0.939 − 0.343i)10-s + 0.365i·11-s + (0.182 + 0.423i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.174 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40\)    =    \(2^{3} \cdot 5\)
Sign: $-0.174 + 0.984i$
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: \((2,\ 40,\ (\ :1),\ -0.174 + 0.984i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.685346 - 0.817632i\)
\(L(\frac12)\) \(\approx\) \(0.685346 - 0.817632i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.376 + 1.96i)T \)
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

−15.41464707505327939104578391050, −14.24682247890604941995045022866, −12.83738684804644557253367181097, −12.03265746395214291190423942213, −11.21247707349386498489225303926, −9.265731312929913675324981149489, −8.534081435595439353902316706645, −5.86050524401683726809087358587, −4.50213454319225510339654013382, −1.71340585576837200706113832598, 4.08477870989734593969642599673, 5.66624847404177367050775249227, 7.25189160141571558037832664134, 8.290880986586959322262782367733, 10.42689097676422202929687231271, 11.06389980171498153397616751204, 13.28504075093105623459820413545, 14.10693900150405982363530706277, 15.05256794949393564274397061208, 16.26136033886806312542870582570

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