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Properties

Label 2-160-20.7-c5-0-20
Degree $2$
Conductor $160$
Sign $0.999 + 0.0151i$
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  + (13.0 + 13.0i)3-s + (−30.1 + 47.1i)5-s + (150. − 150. i)7-s + 97.6i·9-s − 579. i·11-s + (649. − 649. i)13-s + (−1.00e3 + 221. i)15-s + (−106. − 106. i)17-s + 1.20e3·19-s + 3.93e3·21-s + (609. + 609. i)23-s + (−1.31e3 − 2.83e3i)25-s + (1.89e3 − 1.89e3i)27-s + 6.01e3i·29-s + 6.41e3i·31-s + ⋯
L(s)  = 1  + (0.837 + 0.837i)3-s + (−0.538 + 0.842i)5-s + (1.16 − 1.16i)7-s + 0.401i·9-s − 1.44i·11-s + (1.06 − 1.06i)13-s + (−1.15 + 0.254i)15-s + (−0.0893 − 0.0893i)17-s + 0.766·19-s + 1.94·21-s + (0.240 + 0.240i)23-s + (−0.419 − 0.907i)25-s + (0.500 − 0.500i)27-s + 1.32i·29-s + 1.19i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.999 + 0.0151i$
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: \((2,\ 160,\ (\ :5/2),\ 0.999 + 0.0151i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.788060309\)
\(L(\frac12)\) \(\approx\) \(2.788060309\)
\(L(\frac{7}{2})\) 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 \( 1 + (30.1 - 47.1i)T \)
good3 \( 1 + (-13.0 - 13.0i)T + 243iT^{2} \)
7 \( 1 + (-150. + 150. i)T - 1.68e4iT^{2} \)
11 \( 1 + 579. iT - 1.61e5T^{2} \)
13 \( 1 + (-649. + 649. i)T - 3.71e5iT^{2} \)
17 \( 1 + (106. + 106. i)T + 1.41e6iT^{2} \)
19 \( 1 - 1.20e3T + 2.47e6T^{2} \)
23 \( 1 + (-609. - 609. i)T + 6.43e6iT^{2} \)
29 \( 1 - 6.01e3iT - 2.05e7T^{2} \)
31 \( 1 - 6.41e3iT - 2.86e7T^{2} \)
37 \( 1 + (2.85e3 + 2.85e3i)T + 6.93e7iT^{2} \)
41 \( 1 + 1.29e4T + 1.15e8T^{2} \)
43 \( 1 + (3.19e3 + 3.19e3i)T + 1.47e8iT^{2} \)
47 \( 1 + (1.49e4 - 1.49e4i)T - 2.29e8iT^{2} \)
53 \( 1 + (-2.35e4 + 2.35e4i)T - 4.18e8iT^{2} \)
59 \( 1 - 1.69e4T + 7.14e8T^{2} \)
61 \( 1 - 5.12e4T + 8.44e8T^{2} \)
67 \( 1 + (-4.75e3 + 4.75e3i)T - 1.35e9iT^{2} \)
71 \( 1 - 9.19e3iT - 1.80e9T^{2} \)
73 \( 1 + (-2.94e4 + 2.94e4i)T - 2.07e9iT^{2} \)
79 \( 1 + 2.10e4T + 3.07e9T^{2} \)
83 \( 1 + (2.18e4 + 2.18e4i)T + 3.93e9iT^{2} \)
89 \( 1 + 1.02e4iT - 5.58e9T^{2} \)
97 \( 1 + (5.40e4 + 5.40e4i)T + 8.58e9iT^{2} \)
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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

−11.54814762845007492497005729459, −10.82148228522944056610942227329, −10.25052840713561568925447290789, −8.617629282306072160606624153813, −8.104345415379521595658512400449, −6.88259752100944906025993167259, −5.19497644941793795554667391160, −3.66066068414384572125501572191, −3.28086909423186411561953428632, −0.957063696298394575275555005223, 1.45736372450880482479177556704, 2.22267968456416505943591538196, 4.19968099657242295126589882966, 5.30728051732524898834825367776, 6.97951246254048641107406298883, 8.091927833355200313687127883604, 8.567582922001655088303131670389, 9.571513231039177846969857101885, 11.50615728928293502923513818837, 11.94007400814151905614566114004

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