Properties

Label 2-160-5.4-c3-0-17
Degree $2$
Conductor $160$
Sign $-0.998 + 0.0521i$
Analytic cond. $9.44030$
Root an. cond. $3.07250$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.57i·3-s + (−0.582 − 11.1i)5-s − 22.1i·7-s − 46.4·9-s + 27.1·11-s + 70.3i·13-s + (−95.7 + 4.99i)15-s − 73.3i·17-s + 110.·19-s − 189.·21-s + 107. i·23-s + (−124. + 13.0i)25-s + 167. i·27-s − 68.6·29-s − 137.·31-s + ⋯
L(s)  = 1  − 1.64i·3-s + (−0.0521 − 0.998i)5-s − 1.19i·7-s − 1.72·9-s + 0.743·11-s + 1.50i·13-s + (−1.64 + 0.0859i)15-s − 1.04i·17-s + 1.32·19-s − 1.97·21-s + 0.977i·23-s + (−0.994 + 0.104i)25-s + 1.19i·27-s − 0.439·29-s − 0.794·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0521i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0521i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.998 + 0.0521i$
Analytic conductor: \(9.44030\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :3/2),\ -0.998 + 0.0521i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0388225 - 1.48909i\)
\(L(\frac12)\) \(\approx\) \(0.0388225 - 1.48909i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.582 + 11.1i)T \)
good3 \( 1 + 8.57iT - 27T^{2} \)
7 \( 1 + 22.1iT - 343T^{2} \)
11 \( 1 - 27.1T + 1.33e3T^{2} \)
13 \( 1 - 70.3iT - 2.19e3T^{2} \)
17 \( 1 + 73.3iT - 4.91e3T^{2} \)
19 \( 1 - 110.T + 6.85e3T^{2} \)
23 \( 1 - 107. iT - 1.21e4T^{2} \)
29 \( 1 + 68.6T + 2.43e4T^{2} \)
31 \( 1 + 137.T + 2.97e4T^{2} \)
37 \( 1 - 60.3iT - 5.06e4T^{2} \)
41 \( 1 - 95.1T + 6.89e4T^{2} \)
43 \( 1 + 501. iT - 7.95e4T^{2} \)
47 \( 1 + 439. iT - 1.03e5T^{2} \)
53 \( 1 - 286. iT - 1.48e5T^{2} \)
59 \( 1 - 547.T + 2.05e5T^{2} \)
61 \( 1 - 511.T + 2.26e5T^{2} \)
67 \( 1 - 301. iT - 3.00e5T^{2} \)
71 \( 1 - 82.8T + 3.57e5T^{2} \)
73 \( 1 - 763. iT - 3.89e5T^{2} \)
79 \( 1 + 1.01e3T + 4.93e5T^{2} \)
83 \( 1 + 704. iT - 5.71e5T^{2} \)
89 \( 1 - 743.T + 7.04e5T^{2} \)
97 \( 1 + 1.13e3iT - 9.12e5T^{2} \)
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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

−11.79468768450780332416873524534, −11.61311354526994909941417169089, −9.640626254618253328806902630376, −8.714249967145629472395994164130, −7.28899835514460645325450007568, −7.07900258856973387248840941103, −5.48508624881498176015951399284, −3.92580112744876274726015212916, −1.72246242498007187356295720300, −0.74177337154159347003225054451, 2.81761714331742832738234726732, 3.76621696162690482758448157665, 5.29429227994771338051606049269, 6.15327562798036571222126586331, 7.955809438064826697309839414213, 9.111113832517079999205055208127, 9.940053411877276118572797947245, 10.78947480666444603957708240577, 11.57613117909130838297339748789, 12.78431922799145402297042473433

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