Properties

Label 2-160-5.4-c3-0-16
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

\(L(2)\) \(\approx\) \(0.0233895 + 0.897137i\)
\(L(\frac12)\) \(\approx\) \(0.0233895 + 0.897137i\)
\(L(\frac{5}{2})\) 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

−12.10924481357144640119429623653, −10.77775651946370271005761832899, −10.35885014890165992963632355102, −8.220861899250727971507641573924, −7.60598698782053666539761609174, −6.77740362606414402704012015744, −5.75671944528856009079135748962, −3.55167686017964275347178401325, −2.11307239092703246920526119746, −0.40005941588809582139455162648, 2.54259038736404796202911993060, 4.35992245518714077008759898351, 4.91649478382524877285923176531, 6.13944424062498807066843006031, 8.261624042032273344158359608477, 9.108716553478851078774929623137, 9.601469700127525914461069273188, 10.90648166937739243193775782676, 11.79682129153654382675471371661, 12.76714724840683885827473663433

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