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Properties

Label 1-160-160.123-r0-0-0
Degree $1$
Conductor $160$
Sign $0.349 + 0.936i$
Analytic cond. $0.743036$
Root an. cond. $0.743036$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)3-s − 7-s − i·9-s + (0.707 + 0.707i)11-s + (0.707 + 0.707i)13-s − i·17-s + (0.707 − 0.707i)19-s + (−0.707 − 0.707i)21-s − 23-s + (−0.707 + 0.707i)27-s + (0.707 − 0.707i)29-s − 31-s + i·33-s + (0.707 − 0.707i)37-s − i·39-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s − 7-s − i·9-s + (0.707 + 0.707i)11-s + (0.707 + 0.707i)13-s − i·17-s + (0.707 − 0.707i)19-s + (−0.707 − 0.707i)21-s − 23-s + (−0.707 + 0.707i)27-s + (0.707 − 0.707i)29-s − 31-s + i·33-s + (0.707 − 0.707i)37-s − i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(160\)    =    \(2^{5} \cdot 5\)
Sign: $0.349 + 0.936i$
Analytic conductor: \(0.743036\)
Root analytic conductor: \(0.743036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{160} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 160,\ (0:\ ),\ 0.349 + 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.062077183 + 0.7372357847i\)
\(L(\frac12)\) \(\approx\) \(1.062077183 + 0.7372357847i\)
\(L(1)\) \(\approx\) \(1.143206116 + 0.4157001492i\)
\(L(1)\) \(\approx\) \(1.143206116 + 0.4157001492i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 - T \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 - T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 - T \)
37 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 - iT \)
43 \( 1 + (-0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (0.707 + 0.707i)T \)
61 \( 1 + (-0.707 + 0.707i)T \)
67 \( 1 + (-0.707 - 0.707i)T \)
71 \( 1 - iT \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + (-0.707 - 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 - iT \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−27.47334719739164809515984189428, −26.54653972639875947720130889676, −25.433870387292388137845556108066, −25.00484541063223480461264894848, −23.82111263313968997888300051710, −22.83042255835117325233020223618, −21.854130238173875351730677034018, −20.365378243059412437472721903016, −19.88288078816313992206766340609, −18.70527979363781988537460829475, −18.10038748941166004664328566988, −16.55844731339083351149187299843, −15.69907662764061156929408761302, −14.32118254052343372821584148660, −13.58074114596568384530075482615, −12.60272198370992912082448241494, −11.57652408494012876184653893721, −9.97948360740635065242698211265, −9.00725057668705898160390182649, −7.94386974429888065704573111981, −6.73581226509625437845397609747, −5.807029793334279569182567015890, −3.70596733866227766253131146145, −2.89628187763804688810880762934, −1.122550689229588798305487992569, 2.01563360851601363302892106865, 3.50254373935473357422436107773, 4.31398187735061080551003820561, 5.9793579411351496545486763024, 7.215095543501523376571152706522, 8.659994330569323770851859445783, 9.49490617922458695051250701125, 10.37100608988179829751748329075, 11.74398336458579983283811639288, 13.06488131442679288931102630768, 14.00141224713216102388569122554, 15.074713097669053030110477987222, 15.992521092315292336377212474062, 16.806083527930749328664149672025, 18.2257625994392770675077655925, 19.58334481416677322795427778564, 19.9012671700051777605627443592, 21.227542419098082499775190733886, 22.05220919985073606005140761186, 22.93630816886087236557401235270, 24.228660982529641078828052370195, 25.497491061845004824704715935862, 25.94747868914975181355295426631, 26.85871369382040533390031553604, 28.108951789229597143316322049008

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