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Properties

Label 2-160-5.4-c5-0-10
Degree $2$
Conductor $160$
Sign $0.544 - 0.838i$
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  + 19.0i·3-s + (−46.8 − 30.4i)5-s − 91.0i·7-s − 118.·9-s + 290.·11-s − 112. i·13-s + (578. − 891. i)15-s + 939. i·17-s + 2.05e3·19-s + 1.73e3·21-s − 1.68e3i·23-s + (1.27e3 + 2.85e3i)25-s + 2.36e3i·27-s + 3.96e3·29-s − 7.62e3·31-s + ⋯
L(s)  = 1  + 1.21i·3-s + (−0.838 − 0.544i)5-s − 0.702i·7-s − 0.487·9-s + 0.723·11-s − 0.183i·13-s + (0.663 − 1.02i)15-s + 0.788i·17-s + 1.30·19-s + 0.856·21-s − 0.663i·23-s + (0.407 + 0.913i)25-s + 0.625i·27-s + 0.874·29-s − 1.42·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.544 - 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.544 - 0.838i$
Analytic conductor: \(25.6614\)
Root analytic conductor: \(5.06570\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :5/2),\ 0.544 - 0.838i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.694734201\)
\(L(\frac12)\) \(\approx\) \(1.694734201\)
\(L(\frac{7}{2})\) 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 \( 1 + (46.8 + 30.4i)T \)
good3 \( 1 - 19.0iT - 243T^{2} \)
7 \( 1 + 91.0iT - 1.68e4T^{2} \)
11 \( 1 - 290.T + 1.61e5T^{2} \)
13 \( 1 + 112. iT - 3.71e5T^{2} \)
17 \( 1 - 939. iT - 1.41e6T^{2} \)
19 \( 1 - 2.05e3T + 2.47e6T^{2} \)
23 \( 1 + 1.68e3iT - 6.43e6T^{2} \)
29 \( 1 - 3.96e3T + 2.05e7T^{2} \)
31 \( 1 + 7.62e3T + 2.86e7T^{2} \)
37 \( 1 - 9.57e3iT - 6.93e7T^{2} \)
41 \( 1 + 281.T + 1.15e8T^{2} \)
43 \( 1 - 6.51e3iT - 1.47e8T^{2} \)
47 \( 1 + 1.39e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.85e4iT - 4.18e8T^{2} \)
59 \( 1 - 5.19e4T + 7.14e8T^{2} \)
61 \( 1 - 5.15e4T + 8.44e8T^{2} \)
67 \( 1 + 3.90e4iT - 1.35e9T^{2} \)
71 \( 1 - 3.04e4T + 1.80e9T^{2} \)
73 \( 1 - 4.96e4iT - 2.07e9T^{2} \)
79 \( 1 - 9.82e4T + 3.07e9T^{2} \)
83 \( 1 + 9.44e4iT - 3.93e9T^{2} \)
89 \( 1 + 9.82e4T + 5.58e9T^{2} \)
97 \( 1 - 5.90e4iT - 8.58e9T^{2} \)
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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

−11.98013312825504115445098952576, −11.03190781814551632919644621858, −10.13038922125685809508194913084, −9.190163138880815420483515808535, −8.156813474171050972799219977386, −6.94219436482023634784911493585, −5.24439680674311032881913461098, −4.21073935472100982779887088844, −3.49653884561178876254714808338, −0.988460008103250844748982928581, 0.77311123615663630385380667180, 2.28939242844623605100260092962, 3.68614470586510212656225796884, 5.47172574597396474911644716766, 6.85296236486483990150481126107, 7.37015493982048676988884838043, 8.512150698656730591889159757993, 9.677564983043461899292817465068, 11.29834448011704716800307395789, 11.86648034943783250030741319467

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