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Properties

Label 1-140-140.39-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: $\chi_{140} (39, \cdot )$
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

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

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