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Properties

Label 2-160-8.5-c5-0-14
Degree $2$
Conductor $160$
Sign $-0.401 + 0.916i$
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  + 10.8i·3-s + 25i·5-s − 163.·7-s + 125.·9-s + 321. i·11-s − 128. i·13-s − 270.·15-s − 2.11e3·17-s − 1.45e3i·19-s − 1.77e3i·21-s + 1.23e3·23-s − 625·25-s + 3.99e3i·27-s − 4.07e3i·29-s + 3.95e3·31-s + ⋯
L(s)  = 1  + 0.694i·3-s + 0.447i·5-s − 1.26·7-s + 0.517·9-s + 0.801i·11-s − 0.210i·13-s − 0.310·15-s − 1.77·17-s − 0.924i·19-s − 0.876i·21-s + 0.485·23-s − 0.200·25-s + 1.05i·27-s − 0.899i·29-s + 0.739·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.401 + 0.916i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.401 + 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.401 + 0.916i$
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: \((2,\ 160,\ (\ :5/2),\ -0.401 + 0.916i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.1780850552\)
\(L(\frac12)\) \(\approx\) \(0.1780850552\)
\(L(\frac{7}{2})\) 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 \( 1 - 10.8iT - 243T^{2} \)
7 \( 1 + 163.T + 1.68e4T^{2} \)
11 \( 1 - 321. iT - 1.61e5T^{2} \)
13 \( 1 + 128. iT - 3.71e5T^{2} \)
17 \( 1 + 2.11e3T + 1.41e6T^{2} \)
19 \( 1 + 1.45e3iT - 2.47e6T^{2} \)
23 \( 1 - 1.23e3T + 6.43e6T^{2} \)
29 \( 1 + 4.07e3iT - 2.05e7T^{2} \)
31 \( 1 - 3.95e3T + 2.86e7T^{2} \)
37 \( 1 + 1.06e4iT - 6.93e7T^{2} \)
41 \( 1 + 5.90e3T + 1.15e8T^{2} \)
43 \( 1 + 1.64e4iT - 1.47e8T^{2} \)
47 \( 1 + 2.32e4T + 2.29e8T^{2} \)
53 \( 1 + 3.06e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.52e4iT - 7.14e8T^{2} \)
61 \( 1 - 3.91e4iT - 8.44e8T^{2} \)
67 \( 1 - 2.08e4iT - 1.35e9T^{2} \)
71 \( 1 + 1.38e4T + 1.80e9T^{2} \)
73 \( 1 + 4.34e4T + 2.07e9T^{2} \)
79 \( 1 - 1.25e4T + 3.07e9T^{2} \)
83 \( 1 + 6.68e3iT - 3.93e9T^{2} \)
89 \( 1 + 9.04e4T + 5.58e9T^{2} \)
97 \( 1 - 1.49e5T + 8.58e9T^{2} \)
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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

−11.52272076153524987679581165061, −10.45107892454584749567591984315, −9.746746185477138169551501875313, −8.896910652124296911084471798915, −7.15410600053460967147295266704, −6.49484872486966532696045198259, −4.83496652924850145881467725501, −3.75555285938598563877432522905, −2.38905753321191667175676122009, −0.05832455248489167330481723962, 1.41388260377538315760294413664, 3.06362115587212280647553711138, 4.52010860873479644671768918924, 6.20435937241680571504231969024, 6.80842931210754970884380788759, 8.180684210924645157762885771054, 9.191083127010870730370085338681, 10.20476669147752033175655966679, 11.45118957285380750382516222623, 12.56964811871660777359405830949

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