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Properties

Label 1-4851-4851.1532-r0-0-0
Degree $1$
Conductor $4851$
Sign $-0.465 - 0.884i$
Analytic cond. $22.5279$
Root an. cond. $22.5279$
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  + (−0.791 + 0.611i)2-s + (0.251 − 0.967i)4-s + (0.420 − 0.907i)5-s + (0.393 + 0.919i)8-s + (0.222 + 0.974i)10-s + (0.925 + 0.379i)13-s + (−0.873 − 0.486i)16-s + (−0.550 + 0.834i)17-s + (0.809 − 0.587i)19-s + (−0.772 − 0.635i)20-s + (−0.365 − 0.930i)23-s + (−0.646 − 0.762i)25-s + (−0.963 + 0.266i)26-s + (0.842 − 0.538i)29-s + (−0.669 − 0.743i)31-s + (0.988 − 0.149i)32-s + ⋯
L(s)  = 1  + (−0.791 + 0.611i)2-s + (0.251 − 0.967i)4-s + (0.420 − 0.907i)5-s + (0.393 + 0.919i)8-s + (0.222 + 0.974i)10-s + (0.925 + 0.379i)13-s + (−0.873 − 0.486i)16-s + (−0.550 + 0.834i)17-s + (0.809 − 0.587i)19-s + (−0.772 − 0.635i)20-s + (−0.365 − 0.930i)23-s + (−0.646 − 0.762i)25-s + (−0.963 + 0.266i)26-s + (0.842 − 0.538i)29-s + (−0.669 − 0.743i)31-s + (0.988 − 0.149i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4851 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.465 - 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4851\)    =    \(3^{2} \cdot 7^{2} \cdot 11\)
Sign: $-0.465 - 0.884i$
Analytic conductor: \(22.5279\)
Root analytic conductor: \(22.5279\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4851} (1532, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4851,\ (0:\ ),\ -0.465 - 0.884i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3915131700 - 0.6486387487i\)
\(L(\frac12)\) \(\approx\) \(0.3915131700 - 0.6486387487i\)
\(L(1)\) \(\approx\) \(0.7449518191 - 0.04725547602i\)
\(L(1)\) \(\approx\) \(0.7449518191 - 0.04725547602i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.791 + 0.611i)T \)
5 \( 1 + (0.420 - 0.907i)T \)
13 \( 1 + (0.925 + 0.379i)T \)
17 \( 1 + (-0.550 + 0.834i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (-0.365 - 0.930i)T \)
29 \( 1 + (0.842 - 0.538i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.0448 - 0.998i)T \)
41 \( 1 + (0.992 + 0.119i)T \)
43 \( 1 + (-0.733 + 0.680i)T \)
47 \( 1 + (-0.999 + 0.0299i)T \)
53 \( 1 + (0.550 + 0.834i)T \)
59 \( 1 + (-0.599 - 0.800i)T \)
61 \( 1 + (-0.998 - 0.0598i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.936 - 0.351i)T \)
73 \( 1 + (-0.473 - 0.880i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (-0.925 + 0.379i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−18.38447925904801776658940795198, −17.84439609891827049451565591169, −17.33002941888801860138155792278, −16.24304776367804356591943674280, −15.928550040589111320018065094392, −15.10534800981686661568411549246, −14.13876436824780853195345094937, −13.57888086079491456569729010535, −13.00762558926754957393688966021, −11.92721892732074334801696559156, −11.55293419980491926392636412105, −10.74907349751969245836566190030, −10.27698644200888160287737212477, −9.59524458318990573337333948860, −8.93050380040539150480239189204, −8.1551222857208264024274027825, −7.38565293899619650281294616002, −6.83269845912946074629480080386, −6.04027769735093628462120715511, −5.1801338979778818802165555790, −4.04182020460800335168063951596, −3.204461075686748034678086069311, −2.88979988194562947505036198210, −1.76745537096997051290363442901, −1.21300949966882299403492044311, 0.26978797243402846385296481521, 1.272477044875059419055710451552, 1.85952739433456423205102268819, 2.82575056832117175275133488691, 4.22521668664995037021220878073, 4.63451199255924160997251718470, 5.76409817536121503460727181516, 6.04487588329660335515617918664, 6.863950844724053324700840771488, 7.77980993113672831867251672466, 8.425396905212967918599863243474, 8.95977531377508059678390483705, 9.530885538378767559421980290652, 10.278319967808285663157071709802, 11.02323326390652316477231408768, 11.65464152510957458560745460022, 12.616778166478338214989891990734, 13.325302539323079671002822334489, 13.92032092400951247005317401101, 14.63551525790911626606837381775, 15.4786389219737615766090013597, 16.085729569016562295137287411134, 16.50985509612643438027186182730, 17.21372266503289253053104990756, 17.92081831566947820777486100804

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