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Properties

Label 2-160-8.5-c7-0-6
Degree $2$
Conductor $160$
Sign $-0.294 - 0.955i$
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  + 4.99i·3-s + 125i·5-s − 514.·7-s + 2.16e3·9-s + 2.38e3i·11-s − 3.80e3i·13-s − 624.·15-s + 9.79e3·17-s − 4.55e3i·19-s − 2.57e3i·21-s + 1.98e4·23-s − 1.56e4·25-s + 2.17e4i·27-s + 1.41e5i·29-s − 3.58e4·31-s + ⋯
L(s)  = 1  + 0.106i·3-s + 0.447i·5-s − 0.566·7-s + 0.988·9-s + 0.540i·11-s − 0.480i·13-s − 0.0478·15-s + 0.483·17-s − 0.152i·19-s − 0.0605i·21-s + 0.339·23-s − 0.199·25-s + 0.212i·27-s + 1.07i·29-s − 0.216·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.294 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.294 - 0.955i$
Analytic conductor: \(49.9816\)
Root analytic conductor: \(7.06976\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :7/2),\ -0.294 - 0.955i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.554103531\)
\(L(\frac12)\) \(\approx\) \(1.554103531\)
\(L(\frac{9}{2})\) 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 \( 1 - 125iT \)
good3 \( 1 - 4.99iT - 2.18e3T^{2} \)
7 \( 1 + 514.T + 8.23e5T^{2} \)
11 \( 1 - 2.38e3iT - 1.94e7T^{2} \)
13 \( 1 + 3.80e3iT - 6.27e7T^{2} \)
17 \( 1 - 9.79e3T + 4.10e8T^{2} \)
19 \( 1 + 4.55e3iT - 8.93e8T^{2} \)
23 \( 1 - 1.98e4T + 3.40e9T^{2} \)
29 \( 1 - 1.41e5iT - 1.72e10T^{2} \)
31 \( 1 + 3.58e4T + 2.75e10T^{2} \)
37 \( 1 - 2.18e5iT - 9.49e10T^{2} \)
41 \( 1 + 5.52e5T + 1.94e11T^{2} \)
43 \( 1 + 5.77e5iT - 2.71e11T^{2} \)
47 \( 1 - 1.09e6T + 5.06e11T^{2} \)
53 \( 1 - 1.77e6iT - 1.17e12T^{2} \)
59 \( 1 - 1.29e6iT - 2.48e12T^{2} \)
61 \( 1 - 1.16e6iT - 3.14e12T^{2} \)
67 \( 1 - 4.13e6iT - 6.06e12T^{2} \)
71 \( 1 + 2.88e6T + 9.09e12T^{2} \)
73 \( 1 + 1.02e6T + 1.10e13T^{2} \)
79 \( 1 + 2.76e6T + 1.92e13T^{2} \)
83 \( 1 - 8.58e6iT - 2.71e13T^{2} \)
89 \( 1 + 1.40e5T + 4.42e13T^{2} \)
97 \( 1 + 1.15e7T + 8.07e13T^{2} \)
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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

−12.00258639648196334393412598583, −10.58548166132350506223393650768, −10.03874606386481595533364523839, −8.933425395819778639496377583051, −7.47779244361221325478757375658, −6.76918183354396004967866015711, −5.38983050872189431228567201610, −4.04948718306719650703734098363, −2.84761769125660631305657719527, −1.28053771512341511201874464938, 0.43819974388688562644650765153, 1.75882260935402855253653477697, 3.41730587436499857110834226969, 4.59918318960631023413473574639, 5.94931452742319874151346260812, 7.03078504810960736201274608010, 8.155168611843568817708782752917, 9.339955734986008464989342322820, 10.10470842297087936116024844993, 11.34742436463071079405037744670

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