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Properties

Label 2-160-40.29-c5-0-25
Degree $2$
Conductor $160$
Sign $-0.367 + 0.929i$
Analytic cond. $25.6614$
Root an. cond. $5.06570$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 21.6·3-s + (−36.8 + 42.0i)5-s − 236. i·7-s + 227.·9-s − 192. i·11-s − 975.·13-s + (−798. + 911. i)15-s − 670. i·17-s − 456. i·19-s − 5.12e3i·21-s − 2.02e3i·23-s + (−411. − 3.09e3i)25-s − 343.·27-s + 2.78e3i·29-s + 963.·31-s + ⋯
L(s)  = 1  + 1.39·3-s + (−0.658 + 0.752i)5-s − 1.82i·7-s + 0.934·9-s − 0.480i·11-s − 1.60·13-s + (−0.916 + 1.04i)15-s − 0.563i·17-s − 0.290i·19-s − 2.53i·21-s − 0.799i·23-s + (−0.131 − 0.991i)25-s − 0.0906·27-s + 0.614i·29-s + 0.180·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.367 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.367 + 0.929i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ -0.367 + 0.929i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.776576389\)
\(L(\frac12)\) \(\approx\) \(1.776576389\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (36.8 - 42.0i)T \)
good3 \( 1 - 21.6T + 243T^{2} \)
7 \( 1 + 236. iT - 1.68e4T^{2} \)
11 \( 1 + 192. iT - 1.61e5T^{2} \)
13 \( 1 + 975.T + 3.71e5T^{2} \)
17 \( 1 + 670. iT - 1.41e6T^{2} \)
19 \( 1 + 456. iT - 2.47e6T^{2} \)
23 \( 1 + 2.02e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.78e3iT - 2.05e7T^{2} \)
31 \( 1 - 963.T + 2.86e7T^{2} \)
37 \( 1 - 8.72e3T + 6.93e7T^{2} \)
41 \( 1 + 1.81e3T + 1.15e8T^{2} \)
43 \( 1 - 254.T + 1.47e8T^{2} \)
47 \( 1 + 1.97e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.30e4T + 4.18e8T^{2} \)
59 \( 1 + 2.46e4iT - 7.14e8T^{2} \)
61 \( 1 - 189. iT - 8.44e8T^{2} \)
67 \( 1 + 2.49e4T + 1.35e9T^{2} \)
71 \( 1 - 3.82e4T + 1.80e9T^{2} \)
73 \( 1 - 4.47e4iT - 2.07e9T^{2} \)
79 \( 1 + 6.97e4T + 3.07e9T^{2} \)
83 \( 1 + 4.33e3T + 3.93e9T^{2} \)
89 \( 1 + 5.56e3T + 5.58e9T^{2} \)
97 \( 1 + 9.81e4iT - 8.58e9T^{2} \)
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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

−11.55840742128345159089524090464, −10.47646009302936191736762122375, −9.726470683959232107828530584613, −8.357942852992061566055943457213, −7.43722242066650296853121763991, −6.94676063276103966322407756181, −4.54692072938582655866205724710, −3.53673546988746583642393409265, −2.53945516394549095047029550149, −0.45970426428734286116759414294, 1.96387935325329295667895608564, 2.92009578328347212159335725758, 4.42605628648320242605769222019, 5.65055167136113024284584849179, 7.52376710953289471048299526930, 8.259203774316529164930328839735, 9.144937077679733665084377896537, 9.725887269169859394940324287424, 11.69950098044401177042982205070, 12.34197648602268056805699442344

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