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Properties

Label 2-160-8.3-c4-0-13
Degree $2$
Conductor $160$
Sign $0.563 + 0.825i$
Analytic cond. $16.5391$
Root an. cond. $4.06684$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.2·3-s − 11.1i·5-s − 47.5i·7-s + 150.·9-s − 84.0·11-s − 236. i·13-s − 170. i·15-s + 407.·17-s − 124.·19-s − 724. i·21-s − 198. i·23-s − 125.·25-s + 1.05e3·27-s + 1.31e3i·29-s + 1.24e3i·31-s + ⋯
L(s)  = 1  + 1.69·3-s − 0.447i·5-s − 0.971i·7-s + 1.85·9-s − 0.694·11-s − 1.39i·13-s − 0.756i·15-s + 1.41·17-s − 0.343·19-s − 1.64i·21-s − 0.375i·23-s − 0.200·25-s + 1.45·27-s + 1.56i·29-s + 1.29i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.563 + 0.825i$
Analytic conductor: \(16.5391\)
Root analytic conductor: \(4.06684\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :2),\ 0.563 + 0.825i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.81979 - 1.48887i\)
\(L(\frac12)\) \(\approx\) \(2.81979 - 1.48887i\)
\(L(3)\) 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 \( 1 + 11.1iT \)
good3 \( 1 - 15.2T + 81T^{2} \)
7 \( 1 + 47.5iT - 2.40e3T^{2} \)
11 \( 1 + 84.0T + 1.46e4T^{2} \)
13 \( 1 + 236. iT - 2.85e4T^{2} \)
17 \( 1 - 407.T + 8.35e4T^{2} \)
19 \( 1 + 124.T + 1.30e5T^{2} \)
23 \( 1 + 198. iT - 2.79e5T^{2} \)
29 \( 1 - 1.31e3iT - 7.07e5T^{2} \)
31 \( 1 - 1.24e3iT - 9.23e5T^{2} \)
37 \( 1 + 854. iT - 1.87e6T^{2} \)
41 \( 1 - 3.07e3T + 2.82e6T^{2} \)
43 \( 1 + 1.10e3T + 3.41e6T^{2} \)
47 \( 1 + 284. iT - 4.87e6T^{2} \)
53 \( 1 + 2.96e3iT - 7.89e6T^{2} \)
59 \( 1 + 657.T + 1.21e7T^{2} \)
61 \( 1 - 4.20e3iT - 1.38e7T^{2} \)
67 \( 1 - 2.60e3T + 2.01e7T^{2} \)
71 \( 1 - 4.54e3iT - 2.54e7T^{2} \)
73 \( 1 - 822.T + 2.83e7T^{2} \)
79 \( 1 - 3.04e3iT - 3.89e7T^{2} \)
83 \( 1 + 1.12e4T + 4.74e7T^{2} \)
89 \( 1 + 4.38e3T + 6.27e7T^{2} \)
97 \( 1 - 4.44e3T + 8.85e7T^{2} \)
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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

−12.61970898050435612076535118353, −10.60486923607831566451876028270, −10.00313146189513085215010428738, −8.764307666583623614565002744421, −7.961256428845415222593018065978, −7.25752943456632773854853158579, −5.26543386854645650030328396396, −3.78954566675677205474340707149, −2.83191625875302854734587890185, −1.09527278477892981191534660709, 2.01560084676799809241617362572, 2.90005481620528864281528410772, 4.20270093397125306724210551827, 5.97487581217592451019462935841, 7.47702392880393656789363282611, 8.208276950879684494079987872539, 9.301019596331628750364258368419, 9.888999833567989635309483439165, 11.43268054214140075788009219411, 12.51988787064466894327864516092

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