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Properties

Label 1-4140-4140.83-r0-0-0
Degree $1$
Conductor $4140$
Sign $0.832 + 0.554i$
Analytic cond. $19.2260$
Root an. cond. $19.2260$
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  + (0.945 + 0.327i)7-s + (−0.0475 + 0.998i)11-s + (0.945 − 0.327i)13-s + (0.909 − 0.415i)17-s + (−0.415 + 0.909i)19-s + (0.580 − 0.814i)29-s + (−0.928 − 0.371i)31-s + (0.281 − 0.959i)37-s + (−0.235 + 0.971i)41-s + (0.371 + 0.928i)43-s + (−0.866 + 0.5i)47-s + (0.786 + 0.618i)49-s + (0.755 − 0.654i)53-s + (0.327 + 0.945i)59-s + (0.786 − 0.618i)61-s + ⋯
L(s)  = 1  + (0.945 + 0.327i)7-s + (−0.0475 + 0.998i)11-s + (0.945 − 0.327i)13-s + (0.909 − 0.415i)17-s + (−0.415 + 0.909i)19-s + (0.580 − 0.814i)29-s + (−0.928 − 0.371i)31-s + (0.281 − 0.959i)37-s + (−0.235 + 0.971i)41-s + (0.371 + 0.928i)43-s + (−0.866 + 0.5i)47-s + (0.786 + 0.618i)49-s + (0.755 − 0.654i)53-s + (0.327 + 0.945i)59-s + (0.786 − 0.618i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $0.832 + 0.554i$
Analytic conductor: \(19.2260\)
Root analytic conductor: \(19.2260\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4140,\ (0:\ ),\ 0.832 + 0.554i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.155625933 + 0.6518163362i\)
\(L(\frac12)\) \(\approx\) \(2.155625933 + 0.6518163362i\)
\(L(1)\) \(\approx\) \(1.300302777 + 0.1507586310i\)
\(L(1)\) \(\approx\) \(1.300302777 + 0.1507586310i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (0.945 + 0.327i)T \)
11 \( 1 + (-0.0475 + 0.998i)T \)
13 \( 1 + (0.945 - 0.327i)T \)
17 \( 1 + (0.909 - 0.415i)T \)
19 \( 1 + (-0.415 + 0.909i)T \)
29 \( 1 + (0.580 - 0.814i)T \)
31 \( 1 + (-0.928 - 0.371i)T \)
37 \( 1 + (0.281 - 0.959i)T \)
41 \( 1 + (-0.235 + 0.971i)T \)
43 \( 1 + (0.371 + 0.928i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.755 - 0.654i)T \)
59 \( 1 + (0.327 + 0.945i)T \)
61 \( 1 + (0.786 - 0.618i)T \)
67 \( 1 + (0.998 - 0.0475i)T \)
71 \( 1 + (0.841 - 0.540i)T \)
73 \( 1 + (0.909 + 0.415i)T \)
79 \( 1 + (-0.981 - 0.189i)T \)
83 \( 1 + (0.971 - 0.235i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (-0.690 + 0.723i)T \)
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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

−18.43399891232630799293328582794, −17.66353652227114349400780377596, −16.97176492528984751216182967461, −16.40142879684090878620856521416, −15.67896589114047709616358325654, −14.90001538942309547140126560696, −14.1854196263720712586151022086, −13.72358420510054706754030701713, −12.999087271554343041459957430878, −12.117009058965783093838863115856, −11.37055636718918404256575321094, −10.85026359943384258687707134862, −10.32148189533479686856086960929, −9.20544585817389412253350356547, −8.48444597834323657807598341142, −8.162943316780519592975569128258, −7.11188605964402727019393189232, −6.5086029401292930010252581516, −5.50922710207060387107620741008, −5.06776484860425546541829286143, −3.95388232731262927699543130212, −3.5051271118628734364260226433, −2.42955870070217103800595953393, −1.46835176556757794291786043196, −0.780615389827759952075608081000, 0.96051476836558101618967526721, 1.77107533986342309755406601655, 2.49133625532359083762308525085, 3.56873058931744039525998328283, 4.27439433996491210735301357308, 5.08704980835398969915847006427, 5.74914040949283039302835127159, 6.49259110195187136929191079893, 7.55839121250632867114984348703, 7.97037952254272305562452280479, 8.66468468279859766106590940550, 9.62204394227411431594757578446, 10.13248585235355237678945165303, 11.072918070250305262174269893933, 11.54291632366234056405689362695, 12.40726101940526351047643766903, 12.88044917190848940341356028806, 13.820466299949771063794690384338, 14.59615642447453357213533169202, 14.8918235654756261276099596043, 15.785224390792317272102053001270, 16.423855051798910279376212962625, 17.19261179944394739261917989875, 18.02263754384663591211780152843, 18.23273430969443554353384442460

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