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Properties

Label 2-160-5.4-c3-0-11
Degree $2$
Conductor $160$
Sign $0.178 + 0.983i$
Analytic cond. $9.44030$
Root an. cond. $3.07250$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11 + 2i)5-s + 27·9-s − 92i·13-s − 104i·17-s + (117 − 44i)25-s − 130·29-s − 396i·37-s + 230·41-s + (−297 + 54i)45-s + 343·49-s + 572i·53-s − 830·61-s + (184 + 1.01e3i)65-s + 592i·73-s + 729·81-s + ⋯
L(s)  = 1  + (−0.983 + 0.178i)5-s + 9-s − 1.96i·13-s − 1.48i·17-s + (0.936 − 0.351i)25-s − 0.832·29-s − 1.75i·37-s + 0.876·41-s + (−0.983 + 0.178i)45-s + 49-s + 1.48i·53-s − 1.74·61-s + (0.351 + 1.93i)65-s + 0.949i·73-s + 0.999·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.178 + 0.983i$
Analytic conductor: \(9.44030\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :3/2),\ 0.178 + 0.983i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.931654 - 0.777537i\)
\(L(\frac12)\) \(\approx\) \(0.931654 - 0.777537i\)
\(L(\frac{5}{2})\) 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 \( 1 + (11 - 2i)T \)
good3 \( 1 - 27T^{2} \)
7 \( 1 - 343T^{2} \)
11 \( 1 + 1.33e3T^{2} \)
13 \( 1 + 92iT - 2.19e3T^{2} \)
17 \( 1 + 104iT - 4.91e3T^{2} \)
19 \( 1 + 6.85e3T^{2} \)
23 \( 1 - 1.21e4T^{2} \)
29 \( 1 + 130T + 2.43e4T^{2} \)
31 \( 1 + 2.97e4T^{2} \)
37 \( 1 + 396iT - 5.06e4T^{2} \)
41 \( 1 - 230T + 6.89e4T^{2} \)
43 \( 1 - 7.95e4T^{2} \)
47 \( 1 - 1.03e5T^{2} \)
53 \( 1 - 572iT - 1.48e5T^{2} \)
59 \( 1 + 2.05e5T^{2} \)
61 \( 1 + 830T + 2.26e5T^{2} \)
67 \( 1 - 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 592iT - 3.89e5T^{2} \)
79 \( 1 + 4.93e5T^{2} \)
83 \( 1 - 5.71e5T^{2} \)
89 \( 1 + 1.67e3T + 7.04e5T^{2} \)
97 \( 1 + 1.81e3iT - 9.12e5T^{2} \)
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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

−12.33438326636382023820350981237, −11.13828778032039690429166899346, −10.32965164092662315422138116212, −9.135796436023538109969097023523, −7.72137854563861520887544976189, −7.27035454540997435946165121041, −5.56107479388189344856367190790, −4.24929507049613099274517542243, −2.94377924312846552352482580074, −0.61113760722332229735968895935, 1.62138251422118657619404990248, 3.81647198083376130530739013199, 4.58320432736613966477524676957, 6.42903827211856340155560119709, 7.38132275662760033118377177323, 8.498110829954472867085482213523, 9.539483853163395076468002903218, 10.75882802284608102179476389221, 11.72079607174343008749620411088, 12.52933640257135710953824103606

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