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Properties

Label 1-140-140.23-r0-0-0
Degree $1$
Conductor $140$
Sign $0.848 + 0.528i$
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  + (0.866 + 0.5i)3-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + i·13-s + (0.866 + 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)23-s − i·27-s − 29-s + (0.5 − 0.866i)31-s + (0.866 − 0.5i)33-s + (−0.866 + 0.5i)37-s + (−0.5 + 0.866i)39-s + 41-s − i·43-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s + i·13-s + (0.866 + 0.5i)17-s + (−0.5 − 0.866i)19-s + (−0.866 + 0.5i)23-s − i·27-s − 29-s + (0.5 − 0.866i)31-s + (0.866 − 0.5i)33-s + (−0.866 + 0.5i)37-s + (−0.5 + 0.866i)39-s + 41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.848 + 0.528i$
Analytic conductor: \(0.650157\)
Root analytic conductor: \(0.650157\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 140,\ (0:\ ),\ 0.848 + 0.528i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.385217303 + 0.3960753559i\)
\(L(\frac12)\) \(\approx\) \(1.385217303 + 0.3960753559i\)
\(L(1)\) \(\approx\) \(1.327927624 + 0.2388625425i\)
\(L(1)\) \(\approx\) \(1.327927624 + 0.2388625425i\)

Euler product

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

−28.180485228995679476911313495954, −27.30221992214981083581435722805, −26.14389441529036781162818509796, −25.248188498557833230401135491627, −24.666627823005425806670329428503, −23.36077668247142807339684483516, −22.50031918066242097024280813343, −20.9885189894072043463141945983, −20.30897620909083391325087936462, −19.339584005507714935693045865488, −18.32379908130281699147246764210, −17.39913827230440239598917399465, −15.95936730060488392351814690252, −14.79278640534152823281521201908, −14.12910938818513577893782730760, −12.759499771606378662538295663450, −12.137861861437579843461215127076, −10.365282306507836785680133865, −9.372946454647406339523081988746, −8.120039075519840539211158977247, −7.2701559607384545802982506530, −5.908640212008882740294113860950, −4.19067087373393930216152454026, −2.91670792961014948124687092132, −1.517937508265203694735105523809, 1.87676983866037324454985665082, 3.37534943067149906036622323787, 4.37447746055742660511881258651, 5.95505225174585822547768330699, 7.44387115016632638774236229652, 8.63328642097898786322693179877, 9.460953280117753803417157307874, 10.68121004134374144522551540455, 11.862104738805818526803221396773, 13.38567989692758842679060260487, 14.15839400117252671761745725612, 15.152146792513493599737006673947, 16.24735451804053156861652107816, 17.139215183105838896619413169693, 18.854113233694743179133391856759, 19.3542732200895861146096893624, 20.555756079854235590909333199782, 21.50370200887700488241456872333, 22.170305392265966489719082105, 23.7708505501936948607595177581, 24.533713673030609787325079705404, 25.81729846507670701006598835139, 26.30833655722225714151901888471, 27.46761504692350108207954036327, 28.213333120199674031857350302380

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