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Properties

Label 2-160-8.5-c5-0-4
Degree $2$
Conductor $160$
Sign $-0.804 - 0.594i$
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  + 6.93i·3-s + 25i·5-s − 47.1·7-s + 194.·9-s − 253. i·11-s + 1.03e3i·13-s − 173.·15-s + 756.·17-s + 344. i·19-s − 326. i·21-s − 4.97e3·23-s − 625·25-s + 3.03e3i·27-s − 372. i·29-s + 134.·31-s + ⋯
L(s)  = 1  + 0.444i·3-s + 0.447i·5-s − 0.363·7-s + 0.802·9-s − 0.632i·11-s + 1.69i·13-s − 0.198·15-s + 0.634·17-s + 0.218i·19-s − 0.161i·21-s − 1.96·23-s − 0.200·25-s + 0.801i·27-s − 0.0821i·29-s + 0.0251·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 - 0.594i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.804 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.804 - 0.594i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ -0.804 - 0.594i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.229460868\)
\(L(\frac12)\) \(\approx\) \(1.229460868\)
\(L(\frac{7}{2})\) 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 \( 1 - 25iT \)
good3 \( 1 - 6.93iT - 243T^{2} \)
7 \( 1 + 47.1T + 1.68e4T^{2} \)
11 \( 1 + 253. iT - 1.61e5T^{2} \)
13 \( 1 - 1.03e3iT - 3.71e5T^{2} \)
17 \( 1 - 756.T + 1.41e6T^{2} \)
19 \( 1 - 344. iT - 2.47e6T^{2} \)
23 \( 1 + 4.97e3T + 6.43e6T^{2} \)
29 \( 1 + 372. iT - 2.05e7T^{2} \)
31 \( 1 - 134.T + 2.86e7T^{2} \)
37 \( 1 - 6.65e3iT - 6.93e7T^{2} \)
41 \( 1 + 1.59e4T + 1.15e8T^{2} \)
43 \( 1 - 4.77e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.40e4T + 2.29e8T^{2} \)
53 \( 1 - 5.89e3iT - 4.18e8T^{2} \)
59 \( 1 - 3.01e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.31e4iT - 8.44e8T^{2} \)
67 \( 1 + 2.26e4iT - 1.35e9T^{2} \)
71 \( 1 - 5.39e4T + 1.80e9T^{2} \)
73 \( 1 + 5.12e4T + 2.07e9T^{2} \)
79 \( 1 + 4.08e4T + 3.07e9T^{2} \)
83 \( 1 - 1.08e5iT - 3.93e9T^{2} \)
89 \( 1 - 8.19e4T + 5.58e9T^{2} \)
97 \( 1 - 5.25e4T + 8.58e9T^{2} \)
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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

−12.21490278392120679271038786067, −11.40617528227134743922405889741, −10.16918828500970111212457710975, −9.607657221284303922661489546293, −8.297970617647826465883325633675, −7.00588577145368220758295667175, −6.06202835723994198014943708058, −4.46939801854770512075918792877, −3.45642671580976004152417615472, −1.71856443665869784807341483005, 0.39726206232649474017212139552, 1.86067294221119554808503147766, 3.55669501797611434698669195355, 5.00087837280139320209879313558, 6.21348585933407377349336655139, 7.49046809213641953844801018677, 8.240805533502180387377734931192, 9.785702580521614149866916765172, 10.29474614536017237718277411261, 11.89523865599852825184822424815

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