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Properties

Label 1-140-140.59-r0-0-0
Degree $1$
Conductor $140$
Sign $0.386 + 0.922i$
Analytic cond. $0.650157$
Root an. cond. $0.650157$
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  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + 13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s − 27-s + 29-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s − 41-s + 43-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + 13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−0.5 + 0.866i)23-s − 27-s + 29-s + (−0.5 − 0.866i)31-s + (−0.5 + 0.866i)33-s + (0.5 − 0.866i)37-s + (0.5 + 0.866i)39-s − 41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(0.650157\)
Root analytic conductor: \(0.650157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 140,\ (0:\ ),\ 0.386 + 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.050441272 + 0.6987345111i\)
\(L(\frac12)\) \(\approx\) \(1.050441272 + 0.6987345111i\)
\(L(1)\) \(\approx\) \(1.130625634 + 0.4205761847i\)
\(L(1)\) \(\approx\) \(1.130625634 + 0.4205761847i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + T \)
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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

−28.4062201209376822408660781809, −27.09761403208873265642524184472, −26.04286962809984807832842874933, −25.29486273574085971405050111483, −24.14930274742054258245632775972, −23.61497593749107748755689170950, −22.24904891984998054059443102297, −21.148039531688388410896423209881, −19.98213821666575694902464981914, −19.22012861835159759306268601947, −18.24365896672160796085758139991, −17.27581546461861234086279143027, −15.97929899390042034623144567307, −14.73687690726074309569331887906, −13.74631803443243011273789457540, −12.927589215080430250292192407307, −11.7358965771728026605908074175, −10.6180812266695137786460081043, −8.847099780764757615213952500021, −8.36035386516386873813249632474, −6.79453552609884593771343494821, −6.01622858881066360915910831789, −4.06188872131594645528835211712, −2.7483180463612100660989985958, −1.229083646216325875987260333387, 2.03007418941382637973612660122, 3.58718840229542491779446534253, 4.55556284396891775540697969526, 5.98186978004138444129113446815, 7.534522737325870999994964418108, 8.77578558465282088171942015869, 9.68359961255737054561711441533, 10.75861902516497743612646756428, 11.91872094498338069366213651633, 13.40002310777194960914478186783, 14.35342799570736013734243492487, 15.39123513199058973478572264876, 16.20841093209554570556651445150, 17.36540774721001415836688007485, 18.57491307044376305880383911511, 19.865230687495493383838074369866, 20.54118653260363807297261461696, 21.52421584467721632267803655767, 22.55182542796229594942597243704, 23.425291327928134462589285125747, 25.04473987564735285844182217613, 25.54709107821625982568852861261, 26.654107854702306012285977939154, 27.6009896212292958556359824312, 28.2612964548112870948064001194

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