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Properties

Label 1-4851-4851.1732-r1-0-0
Degree $1$
Conductor $4851$
Sign $0.971 + 0.235i$
Analytic cond. $521.312$
Root an. cond. $521.312$
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  + (0.251 + 0.967i)2-s + (−0.873 + 0.486i)4-s + (−0.983 − 0.178i)5-s + (−0.691 − 0.722i)8-s + (−0.0747 − 0.997i)10-s + (−0.251 − 0.967i)13-s + (0.525 − 0.850i)16-s + (−0.992 + 0.119i)17-s + (0.978 + 0.207i)19-s + (0.946 − 0.323i)20-s + (−0.222 + 0.974i)23-s + (0.936 + 0.351i)25-s + (0.873 − 0.486i)26-s + (0.575 − 0.817i)29-s + (0.104 + 0.994i)31-s + (0.955 + 0.294i)32-s + ⋯
L(s)  = 1  + (0.251 + 0.967i)2-s + (−0.873 + 0.486i)4-s + (−0.983 − 0.178i)5-s + (−0.691 − 0.722i)8-s + (−0.0747 − 0.997i)10-s + (−0.251 − 0.967i)13-s + (0.525 − 0.850i)16-s + (−0.992 + 0.119i)17-s + (0.978 + 0.207i)19-s + (0.946 − 0.323i)20-s + (−0.222 + 0.974i)23-s + (0.936 + 0.351i)25-s + (0.873 − 0.486i)26-s + (0.575 − 0.817i)29-s + (0.104 + 0.994i)31-s + (0.955 + 0.294i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.971 + 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4851\)    =    \(3^{2} \cdot 7^{2} \cdot 11\)
Sign: $0.971 + 0.235i$
Analytic conductor: \(521.312\)
Root analytic conductor: \(521.312\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4851} (1732, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4851,\ (1:\ ),\ 0.971 + 0.235i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.104153873 + 0.1318877856i\)
\(L(\frac12)\) \(\approx\) \(1.104153873 + 0.1318877856i\)
\(L(1)\) \(\approx\) \(0.7571051425 + 0.3469791509i\)
\(L(1)\) \(\approx\) \(0.7571051425 + 0.3469791509i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.251 + 0.967i)T \)
5 \( 1 + (-0.983 - 0.178i)T \)
13 \( 1 + (-0.251 - 0.967i)T \)
17 \( 1 + (-0.992 + 0.119i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (0.575 - 0.817i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.575 - 0.817i)T \)
41 \( 1 + (0.280 - 0.959i)T \)
43 \( 1 + (0.826 - 0.563i)T \)
47 \( 1 + (-0.998 - 0.0598i)T \)
53 \( 1 + (-0.599 + 0.800i)T \)
59 \( 1 + (0.280 + 0.959i)T \)
61 \( 1 + (-0.992 + 0.119i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.753 - 0.657i)T \)
73 \( 1 + (0.447 - 0.894i)T \)
79 \( 1 + (0.913 + 0.406i)T \)
83 \( 1 + (-0.251 + 0.967i)T \)
89 \( 1 + (-0.365 - 0.930i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−18.32438422509913580720049799578, −17.41844485125653489298744631436, −16.48944248525794127409644000806, −15.91430426741096451493521838703, −15.07941608401971348121909724308, −14.47090369962985062231581576277, −13.90088904863276041307568446453, −13.05327321233351187011298500593, −12.4628801368792659730075539127, −11.68435154889623417317298421353, −11.32772325450227472614812537714, −10.73997832056954276912069089643, −9.75242660067896459953990719923, −9.27134391443717017774443793671, −8.40936999116473326296161142933, −7.8284439767403366635304326558, −6.77348158023761543558129170667, −6.25048459960806574110703681625, −4.91351036883795563236201288119, −4.60628241953691781125820174339, −3.873992778857164100866308735716, −3.025916499554240316650954736352, −2.434146402882071248686219804080, −1.42662208145873924278206092809, −0.516240024295008769376640088435, 0.28596584422599706323921493512, 1.11868572044784793666314503267, 2.613758104206290193277055626590, 3.4151642534193992582044984452, 4.05551502165700590817169807243, 4.80380582761858629479351623737, 5.46816414240365933359132104962, 6.200754624704287484590750585410, 7.15824153279426259526408556192, 7.5999823369204404612277612933, 8.190344479172302627478551631761, 8.931908313933182911201816389191, 9.58225533626334210093852764047, 10.52994687273852564429665368480, 11.31590428588099015509174110946, 12.22816289347642635579722866127, 12.50140175659863198754483602473, 13.4982920462499842345842364514, 13.91788708588115664375523605036, 14.90579568157524416330758402193, 15.36366529822281465510741125749, 15.90881653014419981631517359692, 16.35932204801418071035982625408, 17.375312144045361920937680374453, 17.761031955296798397499890163767

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