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Properties

Label 2-160-40.29-c3-0-12
Degree $2$
Conductor $160$
Sign $0.810 + 0.585i$
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  + 8.63·3-s + (−2.63 − 10.8i)5-s + 4.74i·7-s + 47.4·9-s − 31.8i·11-s + 59.2·13-s + (−22.7 − 93.7i)15-s − 80.9i·17-s + 114. i·19-s + 40.9i·21-s − 28.3i·23-s + (−111. + 57.3i)25-s + 176.·27-s + 146. i·29-s − 77.6·31-s + ⋯
L(s)  = 1  + 1.66·3-s + (−0.235 − 0.971i)5-s + 0.256i·7-s + 1.75·9-s − 0.873i·11-s + 1.26·13-s + (−0.391 − 1.61i)15-s − 1.15i·17-s + 1.38i·19-s + 0.425i·21-s − 0.257i·23-s + (−0.888 + 0.458i)25-s + 1.25·27-s + 0.937i·29-s − 0.450·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.810 + 0.585i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.810 + 0.585i$
Analytic conductor: \(9.44030\)
Root analytic conductor: \(3.07250\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :3/2),\ 0.810 + 0.585i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.71207 - 0.877813i\)
\(L(\frac12)\) \(\approx\) \(2.71207 - 0.877813i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.63 + 10.8i)T \)
good3 \( 1 - 8.63T + 27T^{2} \)
7 \( 1 - 4.74iT - 343T^{2} \)
11 \( 1 + 31.8iT - 1.33e3T^{2} \)
13 \( 1 - 59.2T + 2.19e3T^{2} \)
17 \( 1 + 80.9iT - 4.91e3T^{2} \)
19 \( 1 - 114. iT - 6.85e3T^{2} \)
23 \( 1 + 28.3iT - 1.21e4T^{2} \)
29 \( 1 - 146. iT - 2.43e4T^{2} \)
31 \( 1 + 77.6T + 2.97e4T^{2} \)
37 \( 1 - 201.T + 5.06e4T^{2} \)
41 \( 1 + 99.3T + 6.89e4T^{2} \)
43 \( 1 + 210.T + 7.95e4T^{2} \)
47 \( 1 - 33.3iT - 1.03e5T^{2} \)
53 \( 1 + 255.T + 1.48e5T^{2} \)
59 \( 1 - 589. iT - 2.05e5T^{2} \)
61 \( 1 - 573. iT - 2.26e5T^{2} \)
67 \( 1 + 23.9T + 3.00e5T^{2} \)
71 \( 1 + 470.T + 3.57e5T^{2} \)
73 \( 1 - 676. iT - 3.89e5T^{2} \)
79 \( 1 + 1.08e3T + 4.93e5T^{2} \)
83 \( 1 - 78.2T + 5.71e5T^{2} \)
89 \( 1 + 606.T + 7.04e5T^{2} \)
97 \( 1 + 151. iT - 9.12e5T^{2} \)
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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

−12.61222051838260663759009398899, −11.42251589063602080812273202126, −9.958892270351004702479195281270, −8.796315863692694197882154697727, −8.558095237903289429478778452621, −7.47315809183704778550336595785, −5.73207537427235327568447763289, −4.11651291656476236998233797427, −3.08096051426246882817860722892, −1.37387895416878819948781250602, 1.98331457401865621233843912372, 3.25929489317259549603954959285, 4.20025029075514045577128039179, 6.44408253620893501065355380649, 7.48060241526015433397519338114, 8.326525012842592133716231876486, 9.374653529039726245686510655186, 10.35707761067524468146430286539, 11.36857266542444763604030178254, 12.94625525786699561633905227281

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