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Properties

Label 2-160-40.29-c7-0-19
Degree $2$
Conductor $160$
Sign $-0.0166 + 0.999i$
Analytic cond. $49.9816$
Root an. cond. $7.06976$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 70.5·3-s + (−35.7 + 277. i)5-s − 1.25e3i·7-s + 2.78e3·9-s + 3.38e3i·11-s − 8.63e3·13-s + (2.51e3 − 1.95e4i)15-s + 3.50e4i·17-s + 1.66e4i·19-s + 8.85e4i·21-s + 6.42e3i·23-s + (−7.55e4 − 1.98e4i)25-s − 4.20e4·27-s + 1.71e5i·29-s − 1.90e5·31-s + ⋯
L(s)  = 1  − 1.50·3-s + (−0.127 + 0.991i)5-s − 1.38i·7-s + 1.27·9-s + 0.766i·11-s − 1.09·13-s + (0.192 − 1.49i)15-s + 1.73i·17-s + 0.556i·19-s + 2.08i·21-s + 0.110i·23-s + (−0.967 − 0.253i)25-s − 0.411·27-s + 1.30i·29-s − 1.14·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0166 + 0.999i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.0166 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $-0.0166 + 0.999i$
Analytic conductor: \(49.9816\)
Root analytic conductor: \(7.06976\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 160,\ (\ :7/2),\ -0.0166 + 0.999i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.2203682185\)
\(L(\frac12)\) \(\approx\) \(0.2203682185\)
\(L(\frac{9}{2})\) 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 \( 1 + (35.7 - 277. i)T \)
good3 \( 1 + 70.5T + 2.18e3T^{2} \)
7 \( 1 + 1.25e3iT - 8.23e5T^{2} \)
11 \( 1 - 3.38e3iT - 1.94e7T^{2} \)
13 \( 1 + 8.63e3T + 6.27e7T^{2} \)
17 \( 1 - 3.50e4iT - 4.10e8T^{2} \)
19 \( 1 - 1.66e4iT - 8.93e8T^{2} \)
23 \( 1 - 6.42e3iT - 3.40e9T^{2} \)
29 \( 1 - 1.71e5iT - 1.72e10T^{2} \)
31 \( 1 + 1.90e5T + 2.75e10T^{2} \)
37 \( 1 - 4.10e5T + 9.49e10T^{2} \)
41 \( 1 - 2.57e5T + 1.94e11T^{2} \)
43 \( 1 + 5.17e5T + 2.71e11T^{2} \)
47 \( 1 + 4.01e5iT - 5.06e11T^{2} \)
53 \( 1 + 7.40e5T + 1.17e12T^{2} \)
59 \( 1 - 2.98e5iT - 2.48e12T^{2} \)
61 \( 1 + 2.77e6iT - 3.14e12T^{2} \)
67 \( 1 + 2.24e6T + 6.06e12T^{2} \)
71 \( 1 + 2.32e4T + 9.09e12T^{2} \)
73 \( 1 + 1.84e6iT - 1.10e13T^{2} \)
79 \( 1 - 7.91e5T + 1.92e13T^{2} \)
83 \( 1 - 1.35e6T + 2.71e13T^{2} \)
89 \( 1 - 1.05e7T + 4.42e13T^{2} \)
97 \( 1 + 1.25e7iT - 8.07e13T^{2} \)
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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

−11.07317229898944127355398464454, −10.53646522065273941908815679759, −9.870873222586881322563019682063, −7.69583736475152650881685481483, −6.97594628174563939998248856131, −6.09379432313335016507911904137, −4.75975107221573779836351412478, −3.68417240224247079394328972993, −1.65851715881765434327239938493, −0.10705938295718826367995450798, 0.74950614703444813578670944330, 2.52863432411961466201481949051, 4.67672140175468691297972147905, 5.33930924756374346926791486137, 6.13358044554347121912539540500, 7.57322648649932830966166624801, 8.961625262755927646660366288461, 9.707316413409432418860286629742, 11.29812686231495910490955476837, 11.76969404313169092201256874871

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