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Properties

Label 2-160-40.19-c4-0-3
Degree $2$
Conductor $160$
Sign $0.256 - 0.966i$
Analytic cond. $16.5391$
Root an. cond. $4.06684$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 10.5i·3-s + (−24.9 + 1.70i)5-s − 26.0·7-s − 30.0·9-s + 94.0·11-s − 236.·13-s + (18.0 + 262. i)15-s + 398. i·17-s − 98.0·19-s + 274. i·21-s + 490.·23-s + (619. − 85.2i)25-s − 537. i·27-s + 1.45e3i·29-s + 442. i·31-s + ⋯
L(s)  = 1  − 1.17i·3-s + (−0.997 + 0.0683i)5-s − 0.530·7-s − 0.370·9-s + 0.776·11-s − 1.39·13-s + (0.0800 + 1.16i)15-s + 1.37i·17-s − 0.271·19-s + 0.621i·21-s + 0.926·23-s + (0.990 − 0.136i)25-s − 0.736i·27-s + 1.72i·29-s + 0.460i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.256 - 0.966i$
Analytic conductor: \(16.5391\)
Root analytic conductor: \(4.06684\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :2),\ 0.256 - 0.966i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.463039 + 0.356115i\)
\(L(\frac12)\) \(\approx\) \(0.463039 + 0.356115i\)
\(L(3)\) 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 \( 1 + (24.9 - 1.70i)T \)
good3 \( 1 + 10.5iT - 81T^{2} \)
7 \( 1 + 26.0T + 2.40e3T^{2} \)
11 \( 1 - 94.0T + 1.46e4T^{2} \)
13 \( 1 + 236.T + 2.85e4T^{2} \)
17 \( 1 - 398. iT - 8.35e4T^{2} \)
19 \( 1 + 98.0T + 1.30e5T^{2} \)
23 \( 1 - 490.T + 2.79e5T^{2} \)
29 \( 1 - 1.45e3iT - 7.07e5T^{2} \)
31 \( 1 - 442. iT - 9.23e5T^{2} \)
37 \( 1 - 1.21e3T + 1.87e6T^{2} \)
41 \( 1 - 336.T + 2.82e6T^{2} \)
43 \( 1 - 1.73e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.94e3T + 4.87e6T^{2} \)
53 \( 1 + 2.00e3T + 7.89e6T^{2} \)
59 \( 1 + 6.52e3T + 1.21e7T^{2} \)
61 \( 1 - 4.99e3iT - 1.38e7T^{2} \)
67 \( 1 - 5.38e3iT - 2.01e7T^{2} \)
71 \( 1 + 6.60e3iT - 2.54e7T^{2} \)
73 \( 1 + 270. iT - 2.83e7T^{2} \)
79 \( 1 + 1.07e3iT - 3.89e7T^{2} \)
83 \( 1 + 8.97e3iT - 4.74e7T^{2} \)
89 \( 1 + 3.55e3T + 6.27e7T^{2} \)
97 \( 1 - 8.43e3iT - 8.85e7T^{2} \)
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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

−12.59671100483607205454716357022, −11.70761947148504294037014586729, −10.55100761564942250848321394748, −9.174805693537658727160759043705, −8.005303677852058131404659660228, −7.12986057950396900208154328190, −6.38358413737659119532289845706, −4.56812972259706020653449365419, −3.09874432652488663315335688618, −1.37304727610956679034331558212, 0.23840793402052761850912064816, 2.96228146386821394436950936226, 4.18846137726065917426131484508, 4.97380094307882125875982270700, 6.75152598901803664652428676622, 7.81062601656837079462888831173, 9.338427991668426263347252931148, 9.697024922329466916211159748058, 11.04099737893628821142468935835, 11.82104178106900614628281461436

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