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Properties

Label 2-160-40.29-c7-0-23
Degree $2$
Conductor $160$
Sign $0.767 + 0.641i$
Analytic cond. $49.9816$
Root an. cond. $7.06976$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 18.1·3-s + (−5.66 + 279. i)5-s + 646. i·7-s − 1.85e3·9-s − 5.05e3i·11-s − 9.20e3·13-s + (103. − 5.08e3i)15-s − 3.39e4i·17-s + 3.48e4i·19-s − 1.17e4i·21-s + 3.39e4i·23-s + (−7.80e4 − 3.16e3i)25-s + 7.35e4·27-s + 6.50e4i·29-s + 1.46e5·31-s + ⋯
L(s)  = 1  − 0.389·3-s + (−0.0202 + 0.999i)5-s + 0.712i·7-s − 0.848·9-s − 1.14i·11-s − 1.16·13-s + (0.00788 − 0.389i)15-s − 1.67i·17-s + 1.16i·19-s − 0.277i·21-s + 0.582i·23-s + (−0.999 − 0.0405i)25-s + 0.719·27-s + 0.494i·29-s + 0.884·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.767 + 0.641i$
Analytic conductor: \(49.9816\)
Root analytic conductor: \(7.06976\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :7/2),\ 0.767 + 0.641i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.9411875839\)
\(L(\frac12)\) \(\approx\) \(0.9411875839\)
\(L(\frac{9}{2})\) 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 \( 1 + (5.66 - 279. i)T \)
good3 \( 1 + 18.1T + 2.18e3T^{2} \)
7 \( 1 - 646. iT - 8.23e5T^{2} \)
11 \( 1 + 5.05e3iT - 1.94e7T^{2} \)
13 \( 1 + 9.20e3T + 6.27e7T^{2} \)
17 \( 1 + 3.39e4iT - 4.10e8T^{2} \)
19 \( 1 - 3.48e4iT - 8.93e8T^{2} \)
23 \( 1 - 3.39e4iT - 3.40e9T^{2} \)
29 \( 1 - 6.50e4iT - 1.72e10T^{2} \)
31 \( 1 - 1.46e5T + 2.75e10T^{2} \)
37 \( 1 - 2.76e5T + 9.49e10T^{2} \)
41 \( 1 + 8.33e5T + 1.94e11T^{2} \)
43 \( 1 - 6.14e5T + 2.71e11T^{2} \)
47 \( 1 + 9.71e5iT - 5.06e11T^{2} \)
53 \( 1 - 7.61e5T + 1.17e12T^{2} \)
59 \( 1 + 9.82e5iT - 2.48e12T^{2} \)
61 \( 1 - 7.25e5iT - 3.14e12T^{2} \)
67 \( 1 - 1.49e6T + 6.06e12T^{2} \)
71 \( 1 + 1.44e6T + 9.09e12T^{2} \)
73 \( 1 + 2.08e6iT - 1.10e13T^{2} \)
79 \( 1 + 2.81e6T + 1.92e13T^{2} \)
83 \( 1 - 9.50e6T + 2.71e13T^{2} \)
89 \( 1 - 3.46e6T + 4.42e13T^{2} \)
97 \( 1 + 4.51e6iT - 8.07e13T^{2} \)
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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

−11.70707230210587567668545344523, −10.55763800643011560320441659774, −9.518028016081297909949015291943, −8.330683173358182359282179739351, −7.16653038318105676455691460803, −6.00691952387968589857533340717, −5.20222636397793321066024737654, −3.28667780040450689956259652539, −2.44557823820839981349059460629, −0.35175515856693259891753086936, 0.824172071325129411200693355316, 2.33714925862032340563518487593, 4.23931738675579026274749788663, 4.99773856856235339227878229377, 6.31128018092862402370949439871, 7.55571355951990516767729250066, 8.590585431981337404946464771126, 9.711073167613042964779114556219, 10.63847930175501180943459318695, 11.86096715574277926678071687372

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