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Properties

Label 2-160-40.29-c5-0-21
Degree $2$
Conductor $160$
Sign $0.149 + 0.988i$
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  + 21.4·3-s + (−48.3 − 27.9i)5-s − 39.9i·7-s + 216.·9-s + 141. i·11-s + 700.·13-s + (−1.03e3 − 600. i)15-s − 960. i·17-s − 2.20e3i·19-s − 855. i·21-s − 4.49e3i·23-s + (1.55e3 + 2.70e3i)25-s − 568.·27-s − 4.83e3i·29-s + 1.12e3·31-s + ⋯
L(s)  = 1  + 1.37·3-s + (−0.865 − 0.500i)5-s − 0.307i·7-s + 0.890·9-s + 0.352i·11-s + 1.14·13-s + (−1.19 − 0.688i)15-s − 0.805i·17-s − 1.39i·19-s − 0.423i·21-s − 1.77i·23-s + (0.498 + 0.866i)25-s − 0.150·27-s − 1.06i·29-s + 0.209·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.149 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.149 + 0.988i$
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: \((2,\ 160,\ (\ :5/2),\ 0.149 + 0.988i)\)

Particular Values

\(L(3)\) \(\approx\) \(2.448088900\)
\(L(\frac12)\) \(\approx\) \(2.448088900\)
\(L(\frac{7}{2})\) 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 \( 1 + (48.3 + 27.9i)T \)
good3 \( 1 - 21.4T + 243T^{2} \)
7 \( 1 + 39.9iT - 1.68e4T^{2} \)
11 \( 1 - 141. iT - 1.61e5T^{2} \)
13 \( 1 - 700.T + 3.71e5T^{2} \)
17 \( 1 + 960. iT - 1.41e6T^{2} \)
19 \( 1 + 2.20e3iT - 2.47e6T^{2} \)
23 \( 1 + 4.49e3iT - 6.43e6T^{2} \)
29 \( 1 + 4.83e3iT - 2.05e7T^{2} \)
31 \( 1 - 1.12e3T + 2.86e7T^{2} \)
37 \( 1 + 8.77e3T + 6.93e7T^{2} \)
41 \( 1 - 1.15e4T + 1.15e8T^{2} \)
43 \( 1 + 1.73e4T + 1.47e8T^{2} \)
47 \( 1 - 1.51e4iT - 2.29e8T^{2} \)
53 \( 1 - 7.11e3T + 4.18e8T^{2} \)
59 \( 1 + 4.17e4iT - 7.14e8T^{2} \)
61 \( 1 - 2.03e4iT - 8.44e8T^{2} \)
67 \( 1 - 2.37e4T + 1.35e9T^{2} \)
71 \( 1 + 881.T + 1.80e9T^{2} \)
73 \( 1 - 7.05e3iT - 2.07e9T^{2} \)
79 \( 1 - 4.77e4T + 3.07e9T^{2} \)
83 \( 1 + 5.16e4T + 3.93e9T^{2} \)
89 \( 1 - 1.22e5T + 5.58e9T^{2} \)
97 \( 1 - 1.55e5iT - 8.58e9T^{2} \)
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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

−11.81183274528058769108316463808, −10.75144639328379023915318439988, −9.344780572174565600090518219058, −8.621452298314359744121514349339, −7.84435194461608830837174870848, −6.74069216283670703329847421864, −4.73859436029779298707385763060, −3.73009090858647714302191355990, −2.51026471688681744400962145562, −0.70043854521558585354532643463, 1.66534353643663402989959648036, 3.31698348662102994502377009912, 3.77640975433722196591555317731, 5.81953131695587094396431989147, 7.27300604039814979644656643498, 8.255187283924123392632070794218, 8.777491315683524706627943820373, 10.12879700730856722767856028321, 11.19221621621909034944156754972, 12.24577875153041347926831887425

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