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Properties

Label 1-140-140.79-r1-0-0
Degree $1$
Conductor $140$
Sign $0.895 - 0.444i$
Analytic cond. $15.0450$
Root an. cond. $15.0450$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $0.895 - 0.444i$
Analytic conductor: \(15.0450\)
Root analytic conductor: \(15.0450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 140,\ (1:\ ),\ 0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.229769258 - 0.2880873219i\)
\(L(\frac12)\) \(\approx\) \(1.229769258 - 0.2880873219i\)
\(L(1)\) \(\approx\) \(0.9179176903 + 0.05825122617i\)
\(L(1)\) \(\approx\) \(0.9179176903 + 0.05825122617i\)

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

−28.32217068170535798316726402309, −27.46253715713258197908104953327, −26.07989492414691402567330441023, −25.11254964400829890831622138644, −24.245398899794642265871893373431, −23.32191584544366040714492327664, −22.40552065820004719324337322911, −21.43292754559462262358262892999, −19.75509319075477566710095386706, −19.40856821902297492612195086208, −17.806594599151445208384631411, −17.46936297406552273675405703762, −16.19119233804162191500270720886, −14.81932430710253783795734406769, −13.781435036481883449152531527285, −12.51869824343291945011251457552, −11.93897043720559363954924696105, −10.61399905243024226166735741342, −9.3578485820375330869275682365, −7.81943092491959813916746666700, −6.98805412970610673066913676987, −5.76687942104938546445939163606, −4.50179390790983385136162430302, −2.57935278930414215198913484542, −1.192596405430610256987633002, 0.61256963377658586632810554850, 2.887597736159593117923781432072, 4.20761872702382287834338789004, 5.37705321947805319331332543627, 6.483547602651855919680175610146, 8.074964332525867009691847082815, 9.42733852544734789482778155829, 10.230451676608975500249744406834, 11.51002095372669305584811433814, 12.24657942303812419273528984654, 13.97985422754405795709216634828, 14.78843658491887073847503350728, 16.12909812824966990260535214137, 16.69500201690640647194609651417, 17.82587048526873969191203419227, 19.03448375331242514489576005272, 20.25852983062655045983677407058, 21.18679446283771737382931382935, 22.18549852913588391710542448430, 22.82596550872619581826021953954, 24.11584133331251052400094124878, 25.059019472812086041724675833223, 26.46542117356426119810663074121, 27.08638795170023888928204983844, 27.910362320131389164819702181101

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