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Properties

Label 2-160-40.29-c5-0-15
Degree $2$
Conductor $160$
Sign $0.704 - 0.710i$
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  + 10.5·3-s + (52.7 − 18.4i)5-s + 47.9i·7-s − 131.·9-s + 690. i·11-s + 743.·13-s + (557. − 194. i)15-s + 2.04e3i·17-s − 1.44e3i·19-s + 506. i·21-s − 1.43e3i·23-s + (2.44e3 − 1.94e3i)25-s − 3.95e3·27-s + 4.38e3i·29-s + 5.82e3·31-s + ⋯
L(s)  = 1  + 0.677·3-s + (0.943 − 0.330i)5-s + 0.370i·7-s − 0.541·9-s + 1.72i·11-s + 1.21·13-s + (0.639 − 0.223i)15-s + 1.71i·17-s − 0.919i·19-s + 0.250i·21-s − 0.565i·23-s + (0.782 − 0.623i)25-s − 1.04·27-s + 0.968i·29-s + 1.08·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.710i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.704 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.704 - 0.710i$
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: \((2,\ 160,\ (\ :5/2),\ 0.704 - 0.710i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.818362954\)
\(L(\frac12)\) \(\approx\) \(2.818362954\)
\(L(\frac{7}{2})\) 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 \( 1 + (-52.7 + 18.4i)T \)
good3 \( 1 - 10.5T + 243T^{2} \)
7 \( 1 - 47.9iT - 1.68e4T^{2} \)
11 \( 1 - 690. iT - 1.61e5T^{2} \)
13 \( 1 - 743.T + 3.71e5T^{2} \)
17 \( 1 - 2.04e3iT - 1.41e6T^{2} \)
19 \( 1 + 1.44e3iT - 2.47e6T^{2} \)
23 \( 1 + 1.43e3iT - 6.43e6T^{2} \)
29 \( 1 - 4.38e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.82e3T + 2.86e7T^{2} \)
37 \( 1 - 632.T + 6.93e7T^{2} \)
41 \( 1 - 9.13e3T + 1.15e8T^{2} \)
43 \( 1 + 1.99e3T + 1.47e8T^{2} \)
47 \( 1 - 1.29e3iT - 2.29e8T^{2} \)
53 \( 1 + 7.76e3T + 4.18e8T^{2} \)
59 \( 1 - 7.01e3iT - 7.14e8T^{2} \)
61 \( 1 - 5.00e3iT - 8.44e8T^{2} \)
67 \( 1 - 1.45e4T + 1.35e9T^{2} \)
71 \( 1 - 5.97e4T + 1.80e9T^{2} \)
73 \( 1 - 3.77e4iT - 2.07e9T^{2} \)
79 \( 1 + 7.81e4T + 3.07e9T^{2} \)
83 \( 1 + 8.70e3T + 3.93e9T^{2} \)
89 \( 1 + 7.88e4T + 5.58e9T^{2} \)
97 \( 1 + 5.86e3iT - 8.58e9T^{2} \)
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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

−12.42830472248591312545915191645, −10.95967816998138535000478768715, −9.932444407483436461166394725793, −8.944487434117294517062702917962, −8.290491539748901947292839365139, −6.71497499925880312242995067940, −5.65937499455261267178121765615, −4.28261322942107619506535768490, −2.63055529304384697450092101928, −1.55393470325276154854491506545, 0.910425889410618448127791425118, 2.64515134022341935882646297989, 3.58449613766396686814014099325, 5.55331308186773322260929555318, 6.33723927493379707360357108633, 7.87793724179081313020014919222, 8.798217962704078407502459195908, 9.687549735410720881554852447000, 10.89833640663805568715744747562, 11.61954091189445534053827312977

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