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Properties

Label 2-160-20.7-c5-0-23
Degree $2$
Conductor $160$
Sign $-0.346 + 0.937i$
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  + (12.0 + 12.0i)3-s + (−34.4 − 44.0i)5-s + (−74.8 + 74.8i)7-s + 49.4i·9-s + 432. i·11-s + (639. − 639. i)13-s + (116. − 948. i)15-s + (−1.42e3 − 1.42e3i)17-s − 2.37e3·19-s − 1.81e3·21-s + (170. + 170. i)23-s + (−757. + 3.03e3i)25-s + (2.34e3 − 2.34e3i)27-s − 5.00e3i·29-s − 7.16e3i·31-s + ⋯
L(s)  = 1  + (0.775 + 0.775i)3-s + (−0.615 − 0.788i)5-s + (−0.577 + 0.577i)7-s + 0.203i·9-s + 1.07i·11-s + (1.04 − 1.04i)13-s + (0.133 − 1.08i)15-s + (−1.19 − 1.19i)17-s − 1.50·19-s − 0.895·21-s + (0.0671 + 0.0671i)23-s + (−0.242 + 0.970i)25-s + (0.617 − 0.617i)27-s − 1.10i·29-s − 1.33i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.346 + 0.937i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.346 + 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.346 + 0.937i$
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: \((2,\ 160,\ (\ :5/2),\ -0.346 + 0.937i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.8704608064\)
\(L(\frac12)\) \(\approx\) \(0.8704608064\)
\(L(\frac{7}{2})\) 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 \( 1 + (34.4 + 44.0i)T \)
good3 \( 1 + (-12.0 - 12.0i)T + 243iT^{2} \)
7 \( 1 + (74.8 - 74.8i)T - 1.68e4iT^{2} \)
11 \( 1 - 432. iT - 1.61e5T^{2} \)
13 \( 1 + (-639. + 639. i)T - 3.71e5iT^{2} \)
17 \( 1 + (1.42e3 + 1.42e3i)T + 1.41e6iT^{2} \)
19 \( 1 + 2.37e3T + 2.47e6T^{2} \)
23 \( 1 + (-170. - 170. i)T + 6.43e6iT^{2} \)
29 \( 1 + 5.00e3iT - 2.05e7T^{2} \)
31 \( 1 + 7.16e3iT - 2.86e7T^{2} \)
37 \( 1 + (4.64e3 + 4.64e3i)T + 6.93e7iT^{2} \)
41 \( 1 - 1.57e3T + 1.15e8T^{2} \)
43 \( 1 + (4.52e3 + 4.52e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (-489. + 489. i)T - 2.29e8iT^{2} \)
53 \( 1 + (1.53e3 - 1.53e3i)T - 4.18e8iT^{2} \)
59 \( 1 - 2.54e3T + 7.14e8T^{2} \)
61 \( 1 + 3.14e4T + 8.44e8T^{2} \)
67 \( 1 + (3.79e4 - 3.79e4i)T - 1.35e9iT^{2} \)
71 \( 1 + 4.14e4iT - 1.80e9T^{2} \)
73 \( 1 + (-3.76e4 + 3.76e4i)T - 2.07e9iT^{2} \)
79 \( 1 - 3.97e4T + 3.07e9T^{2} \)
83 \( 1 + (4.03e4 + 4.03e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 4.83e4iT - 5.58e9T^{2} \)
97 \( 1 + (-6.01e4 - 6.01e4i)T + 8.58e9iT^{2} \)
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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

−11.76021658755296693971333833334, −10.50169438425168553828492093441, −9.328939166863691959301025968276, −8.849255946706480687416308185331, −7.77791224319098859628143934613, −6.23364363664962838707853897319, −4.69696917287948654460133321260, −3.81946622318301853089893582848, −2.43571397938896933413995454360, −0.24830470077930439217168376090, 1.66502583403857000031525412542, 3.13490356367081216279331344211, 4.10037302990281648563331578764, 6.47390715532336780711256656848, 6.82793887768240113294798426621, 8.328580774215637187844827680967, 8.733560718207595265576043041944, 10.66201491964647488982829459196, 11.01203300048445565557070885843, 12.51761272684607225467596499241

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