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Properties

Label 1-4729-4729.1087-r0-0-0
Degree 11
Conductor 47294729
Sign 0.0921−0.995i0.0921 - 0.995i
Analytic cond. 21.961321.9613
Root an. cond. 21.961321.9613
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.753 + 0.657i)2-s + (0.651 + 0.758i)3-s + (0.135 − 0.990i)4-s + (0.975 − 0.221i)5-s + (−0.989 − 0.143i)6-s + (−0.908 + 0.417i)7-s + (0.549 + 0.835i)8-s + (−0.150 + 0.988i)9-s + (−0.589 + 0.808i)10-s + (−0.698 + 0.715i)11-s + (0.839 − 0.543i)12-s + (−0.978 + 0.205i)13-s + (0.410 − 0.912i)14-s + (0.803 + 0.595i)15-s + (−0.963 − 0.267i)16-s + (−0.999 + 0.0318i)17-s + ⋯
L(s)  = 1  + (−0.753 + 0.657i)2-s + (0.651 + 0.758i)3-s + (0.135 − 0.990i)4-s + (0.975 − 0.221i)5-s + (−0.989 − 0.143i)6-s + (−0.908 + 0.417i)7-s + (0.549 + 0.835i)8-s + (−0.150 + 0.988i)9-s + (−0.589 + 0.808i)10-s + (−0.698 + 0.715i)11-s + (0.839 − 0.543i)12-s + (−0.978 + 0.205i)13-s + (0.410 − 0.912i)14-s + (0.803 + 0.595i)15-s + (−0.963 − 0.267i)16-s + (−0.999 + 0.0318i)17-s + ⋯

Functional equation

Λ(s)=(4729s/2 ΓR(s) L(s)=((0.0921−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0921 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(4729s/2 ΓR(s) L(s)=((0.0921−0.995i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4729 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0921 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 47294729
Sign: 0.0921−0.995i0.0921 - 0.995i
Analytic conductor: 21.961321.9613
Root analytic conductor: 21.961321.9613
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ4729(1087,⋅)\chi_{4729} (1087, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 4729, (0: ), 0.0921−0.995i)(1,\ 4729,\ (0:\ ),\ 0.0921 - 0.995i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx −0.3062119043+0.2791940753i-0.3062119043 + 0.2791940753i
L(12)L(\frac12) ≈\approx −0.3062119043+0.2791940753i-0.3062119043 + 0.2791940753i
L(1)L(1) ≈\approx 0.5500461513+0.5016380510i0.5500461513 + 0.5016380510i
L(1)L(1) ≈\approx 0.5500461513+0.5016380510i0.5500461513 + 0.5016380510i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad4729 1 1
good2 1+(−0.753+0.657i)T 1 + (-0.753 + 0.657i)T
3 1+(0.651+0.758i)T 1 + (0.651 + 0.758i)T
5 1+(0.975−0.221i)T 1 + (0.975 - 0.221i)T
7 1+(−0.908+0.417i)T 1 + (-0.908 + 0.417i)T
11 1+(−0.698+0.715i)T 1 + (-0.698 + 0.715i)T
13 1+(−0.978+0.205i)T 1 + (-0.978 + 0.205i)T
17 1+(−0.999+0.0318i)T 1 + (-0.999 + 0.0318i)T
19 1+(−0.244+0.969i)T 1 + (-0.244 + 0.969i)T
23 1+(−0.467+0.884i)T 1 + (-0.467 + 0.884i)T
29 1+(0.753+0.657i)T 1 + (0.753 + 0.657i)T
31 1+(0.954+0.298i)T 1 + (0.954 + 0.298i)T
37 1+(−0.119−0.992i)T 1 + (-0.119 - 0.992i)T
41 1+(−0.987−0.158i)T 1 + (-0.987 - 0.158i)T
43 1+(−0.166+0.986i)T 1 + (-0.166 + 0.986i)T
47 1+(−0.856+0.516i)T 1 + (-0.856 + 0.516i)T
53 1+(0.812−0.582i)T 1 + (0.812 - 0.582i)T
59 1+(0.103−0.994i)T 1 + (0.103 - 0.994i)T
61 1+(0.709+0.704i)T 1 + (0.709 + 0.704i)T
67 1+(0.439−0.898i)T 1 + (0.439 - 0.898i)T
71 1+(−0.732+0.681i)T 1 + (-0.732 + 0.681i)T
73 1+(0.763+0.645i)T 1 + (0.763 + 0.645i)T
79 1−T 1 - T
83 1+(0.967−0.252i)T 1 + (0.967 - 0.252i)T
89 1+(0.998+0.0478i)T 1 + (0.998 + 0.0478i)T
97 1+(−0.908+0.417i)T 1 + (-0.908 + 0.417i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−17.738926293252233595217286562234, −17.2481173924775847739679692518, −16.61596250844269094012075934514, −15.6633066859006864397078134933, −15.02184403377705805042806638776, −13.82991255521435823310737625775, −13.4471341581854711801538717523, −13.12435051496938101772322198165, −12.28660645430026045964989953547, −11.61904301394369160582256042983, −10.550771571138270234766153540899, −10.14391930852927450625301003477, −9.52853124058271726388446372829, −8.73218262219621528420666872017, −8.28278085788592386124142189648, −7.32639640842806701911844189170, −6.63590612142575799432795162202, −6.3385394783685000206231520420, −5.00411326816399553144163872951, −4.0065340353012248718148173873, −2.90087868713553594648340138380, −2.67974943334925335594223231537, −2.06194932026332229419651791259, −0.89185722320553482664388945567, −0.13813509064418751310416045044, 1.61104604530395893300753149251, 2.27366849887890734026433425026, 2.82764194310773087838330171730, 4.085861861578052562892242846644, 5.05097025270329214044158317311, 5.346098010719941103545057978774, 6.37401773323184195336670124315, 6.91764471680733279979280067555, 7.89062060547594122533213722776, 8.52588007843941079222305869200, 9.25385802362403029622416390671, 9.7903708910684475178329611766, 10.09435284710644864193405038455, 10.73775053041493555951893892027, 11.912117586360282135944233462523, 12.84653342433171755084510452448, 13.45588512581973542208137873601, 14.2126121375992682342889804584, 14.78561046542210573358716647232, 15.434440502221799030853652264684, 16.12314688711100967629513024396, 16.42899352852824293540974779869, 17.475533424900117534072160743294, 17.715267946089878294203971295, 18.67052392220257074774581779771

Graph of the ZZ-function along the critical line