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Properties

Label 2-160-40.29-c5-0-12
Degree $2$
Conductor $160$
Sign $0.961 - 0.273i$
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  − 7.17·3-s + (1.28 + 55.8i)5-s − 146. i·7-s − 191.·9-s − 42.4i·11-s + 605.·13-s + (−9.21 − 401. i)15-s + 409. i·17-s − 2.09e3i·19-s + 1.05e3i·21-s + 3.04e3i·23-s + (−3.12e3 + 143. i)25-s + 3.11e3·27-s + 4.59e3i·29-s + 5.11e3·31-s + ⋯
L(s)  = 1  − 0.460·3-s + (0.0229 + 0.999i)5-s − 1.13i·7-s − 0.787·9-s − 0.105i·11-s + 0.993·13-s + (−0.0105 − 0.460i)15-s + 0.343i·17-s − 1.33i·19-s + 0.521i·21-s + 1.20i·23-s + (−0.998 + 0.0459i)25-s + 0.823·27-s + 1.01i·29-s + 0.956·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.961 - 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.961 - 0.273i$
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: \((2,\ 160,\ (\ :5/2),\ 0.961 - 0.273i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.496404632\)
\(L(\frac12)\) \(\approx\) \(1.496404632\)
\(L(\frac{7}{2})\) 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 \( 1 + (-1.28 - 55.8i)T \)
good3 \( 1 + 7.17T + 243T^{2} \)
7 \( 1 + 146. iT - 1.68e4T^{2} \)
11 \( 1 + 42.4iT - 1.61e5T^{2} \)
13 \( 1 - 605.T + 3.71e5T^{2} \)
17 \( 1 - 409. iT - 1.41e6T^{2} \)
19 \( 1 + 2.09e3iT - 2.47e6T^{2} \)
23 \( 1 - 3.04e3iT - 6.43e6T^{2} \)
29 \( 1 - 4.59e3iT - 2.05e7T^{2} \)
31 \( 1 - 5.11e3T + 2.86e7T^{2} \)
37 \( 1 - 1.12e4T + 6.93e7T^{2} \)
41 \( 1 - 1.10e4T + 1.15e8T^{2} \)
43 \( 1 - 8.41e3T + 1.47e8T^{2} \)
47 \( 1 + 3.97e3iT - 2.29e8T^{2} \)
53 \( 1 - 3.82e4T + 4.18e8T^{2} \)
59 \( 1 - 3.66e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.18e3iT - 8.44e8T^{2} \)
67 \( 1 - 4.18e4T + 1.35e9T^{2} \)
71 \( 1 + 7.21e4T + 1.80e9T^{2} \)
73 \( 1 + 4.82e4iT - 2.07e9T^{2} \)
79 \( 1 - 3.94e4T + 3.07e9T^{2} \)
83 \( 1 + 4.65e4T + 3.93e9T^{2} \)
89 \( 1 + 3.82e4T + 5.58e9T^{2} \)
97 \( 1 + 9.59e4iT - 8.58e9T^{2} \)
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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

−11.64628121816894380286486869917, −11.01999764782800522103555952978, −10.36328839691679418022651117791, −8.987376000892122960498813645190, −7.64065087038369035869462937528, −6.68805837195470221954065644602, −5.70514850685648791478419220713, −4.08124025999037292172780226244, −2.86892139114213569612945741833, −0.857438929080818437913635729471, 0.78223106472522402994549574091, 2.46783801812073663528735592166, 4.27350235008974955661003370362, 5.61560727497995662042227355326, 6.13206802150415068417599365049, 8.136017929767346452409330717702, 8.712364947766025635943118760376, 9.805588422225809974723874648082, 11.18593392179090501561816661019, 12.01456959363760616786659831503

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