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Properties

Label 2-160-20.7-c5-0-29
Degree $2$
Conductor $160$
Sign $0.277 + 0.960i$
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  + (13.4 + 13.4i)3-s + (15.0 − 53.8i)5-s + (76.4 − 76.4i)7-s + 117. i·9-s − 622. i·11-s + (−293. + 293. i)13-s + (925. − 521. i)15-s + (−1.15e3 − 1.15e3i)17-s − 2.00e3·19-s + 2.05e3·21-s + (−1.39e3 − 1.39e3i)23-s + (−2.67e3 − 1.61e3i)25-s + (1.68e3 − 1.68e3i)27-s − 305. i·29-s + 2.10e3i·31-s + ⋯
L(s)  = 1  + (0.861 + 0.861i)3-s + (0.268 − 0.963i)5-s + (0.589 − 0.589i)7-s + 0.485i·9-s − 1.55i·11-s + (−0.481 + 0.481i)13-s + (1.06 − 0.598i)15-s + (−0.967 − 0.967i)17-s − 1.27·19-s + 1.01·21-s + (−0.548 − 0.548i)23-s + (−0.855 − 0.518i)25-s + (0.443 − 0.443i)27-s − 0.0675i·29-s + 0.393i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.277 + 0.960i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.338703766\)
\(L(\frac12)\) \(\approx\) \(2.338703766\)
\(L(\frac{7}{2})\) 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 \( 1 + (-15.0 + 53.8i)T \)
good3 \( 1 + (-13.4 - 13.4i)T + 243iT^{2} \)
7 \( 1 + (-76.4 + 76.4i)T - 1.68e4iT^{2} \)
11 \( 1 + 622. iT - 1.61e5T^{2} \)
13 \( 1 + (293. - 293. i)T - 3.71e5iT^{2} \)
17 \( 1 + (1.15e3 + 1.15e3i)T + 1.41e6iT^{2} \)
19 \( 1 + 2.00e3T + 2.47e6T^{2} \)
23 \( 1 + (1.39e3 + 1.39e3i)T + 6.43e6iT^{2} \)
29 \( 1 + 305. iT - 2.05e7T^{2} \)
31 \( 1 - 2.10e3iT - 2.86e7T^{2} \)
37 \( 1 + (-9.90e3 - 9.90e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.69e4T + 1.15e8T^{2} \)
43 \( 1 + (-2.03e3 - 2.03e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (-682. + 682. i)T - 2.29e8iT^{2} \)
53 \( 1 + (-2.10e4 + 2.10e4i)T - 4.18e8iT^{2} \)
59 \( 1 + 1.16e4T + 7.14e8T^{2} \)
61 \( 1 + 1.30e3T + 8.44e8T^{2} \)
67 \( 1 + (-3.98e4 + 3.98e4i)T - 1.35e9iT^{2} \)
71 \( 1 - 2.54e4iT - 1.80e9T^{2} \)
73 \( 1 + (-1.03e3 + 1.03e3i)T - 2.07e9iT^{2} \)
79 \( 1 - 1.18e4T + 3.07e9T^{2} \)
83 \( 1 + (4.51e4 + 4.51e4i)T + 3.93e9iT^{2} \)
89 \( 1 - 1.43e5iT - 5.58e9T^{2} \)
97 \( 1 + (2.43e4 + 2.43e4i)T + 8.58e9iT^{2} \)
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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

−11.63612077708158549869827067932, −10.67035507117230466723547259312, −9.513746312229115989997681276890, −8.755980231068830018367729419309, −8.057268871138867448995034586715, −6.32328811164269678452302899844, −4.77236481636696110770506840167, −4.05588460442853027570232710896, −2.47227600649394588358038408774, −0.65526603359505334732628623761, 2.03008847402915972302849716039, 2.36421841297253055468180815474, 4.26160443679404478521506102435, 5.96684740140549637304405277370, 7.16221944789467641676787477259, 7.84905374008102032332369063032, 8.972274878650761543337136881649, 10.16290451367350245627494333983, 11.16881288745266491779947598033, 12.51101331257629629942244238557

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