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Properties

Label 2-160-40.29-c5-0-22
Degree $2$
Conductor $160$
Sign $0.771 + 0.635i$
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  + 28.9·3-s + (40.5 − 38.5i)5-s − 128. i·7-s + 592.·9-s + 433. i·11-s − 78.5·13-s + (1.17e3 − 1.11e3i)15-s − 1.48e3i·17-s − 98.3i·19-s − 3.70e3i·21-s + 2.36e3i·23-s + (156. − 3.12e3i)25-s + 1.01e4·27-s + 1.02e3i·29-s − 4.30e3·31-s + ⋯
L(s)  = 1  + 1.85·3-s + (0.724 − 0.689i)5-s − 0.988i·7-s + 2.43·9-s + 1.08i·11-s − 0.128·13-s + (1.34 − 1.27i)15-s − 1.24i·17-s − 0.0624i·19-s − 1.83i·21-s + 0.931i·23-s + (0.0500 − 0.998i)25-s + 2.66·27-s + 0.226i·29-s − 0.805·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 + 0.635i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.771 + 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.771 + 0.635i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ 0.771 + 0.635i)\)

Particular Values

\(L(3)\) \(\approx\) \(4.290041282\)
\(L(\frac12)\) \(\approx\) \(4.290041282\)
\(L(\frac{7}{2})\) 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 \( 1 + (-40.5 + 38.5i)T \)
good3 \( 1 - 28.9T + 243T^{2} \)
7 \( 1 + 128. iT - 1.68e4T^{2} \)
11 \( 1 - 433. iT - 1.61e5T^{2} \)
13 \( 1 + 78.5T + 3.71e5T^{2} \)
17 \( 1 + 1.48e3iT - 1.41e6T^{2} \)
19 \( 1 + 98.3iT - 2.47e6T^{2} \)
23 \( 1 - 2.36e3iT - 6.43e6T^{2} \)
29 \( 1 - 1.02e3iT - 2.05e7T^{2} \)
31 \( 1 + 4.30e3T + 2.86e7T^{2} \)
37 \( 1 + 6.64e3T + 6.93e7T^{2} \)
41 \( 1 + 4.44e3T + 1.15e8T^{2} \)
43 \( 1 - 7.06e3T + 1.47e8T^{2} \)
47 \( 1 - 1.55e4iT - 2.29e8T^{2} \)
53 \( 1 - 2.67e4T + 4.18e8T^{2} \)
59 \( 1 - 3.29e4iT - 7.14e8T^{2} \)
61 \( 1 + 1.92e4iT - 8.44e8T^{2} \)
67 \( 1 + 1.06e3T + 1.35e9T^{2} \)
71 \( 1 + 1.23e4T + 1.80e9T^{2} \)
73 \( 1 + 3.72e4iT - 2.07e9T^{2} \)
79 \( 1 - 5.07e4T + 3.07e9T^{2} \)
83 \( 1 - 3.86e4T + 3.93e9T^{2} \)
89 \( 1 + 1.32e5T + 5.58e9T^{2} \)
97 \( 1 - 9.50e4iT - 8.58e9T^{2} \)
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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

−12.25880879073438379601129711138, −10.38882439085342802174331640835, −9.553498306899705406468648699724, −9.001688832429400082186348366238, −7.68864471426426433650457865541, −7.05923024490043444702975275331, −4.93807627397836203193211492018, −3.84913435055689993034088501341, −2.43392026600423444730794756187, −1.30037215457518147854774132517, 1.85453916118455135690312723066, 2.74191775085184694731895931538, 3.75467442040380176781557879146, 5.71380529088814326493104186044, 6.94585581231756381260895243280, 8.332839204252586703087902316998, 8.808459347227137437806525036289, 9.852508052611344488344255668640, 10.79653367828342635320885326681, 12.42500688376588714493332137948

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