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Properties

Label 1-4760-4760.1573-r0-0-0
Degree $1$
Conductor $4760$
Sign $-0.780 - 0.625i$
Analytic cond. $22.1053$
Root an. cond. $22.1053$
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  + (−0.258 − 0.965i)3-s + (−0.866 + 0.5i)9-s + (−0.258 − 0.965i)11-s − i·13-s + (0.866 − 0.5i)19-s + (0.258 − 0.965i)23-s + (0.707 + 0.707i)27-s + (−0.707 − 0.707i)29-s + (0.965 − 0.258i)31-s + (−0.866 + 0.5i)33-s + (0.965 + 0.258i)37-s + (−0.965 + 0.258i)39-s + (0.707 − 0.707i)41-s + 43-s + (−0.866 + 0.5i)47-s + ⋯
L(s)  = 1  + (−0.258 − 0.965i)3-s + (−0.866 + 0.5i)9-s + (−0.258 − 0.965i)11-s − i·13-s + (0.866 − 0.5i)19-s + (0.258 − 0.965i)23-s + (0.707 + 0.707i)27-s + (−0.707 − 0.707i)29-s + (0.965 − 0.258i)31-s + (−0.866 + 0.5i)33-s + (0.965 + 0.258i)37-s + (−0.965 + 0.258i)39-s + (0.707 − 0.707i)41-s + 43-s + (−0.866 + 0.5i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4760\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 17\)
Sign: $-0.780 - 0.625i$
Analytic conductor: \(22.1053\)
Root analytic conductor: \(22.1053\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4760} (1573, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4760,\ (0:\ ),\ -0.780 - 0.625i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5148738266 - 1.466708860i\)
\(L(\frac12)\) \(\approx\) \(0.5148738266 - 1.466708860i\)
\(L(1)\) \(\approx\) \(0.8616939238 - 0.5220050329i\)
\(L(1)\) \(\approx\) \(0.8616939238 - 0.5220050329i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.258 - 0.965i)T \)
11 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 - iT \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (0.258 - 0.965i)T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + (0.965 - 0.258i)T \)
37 \( 1 + (0.965 + 0.258i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (0.866 + 0.5i)T \)
61 \( 1 + (0.965 + 0.258i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.707 + 0.707i)T \)
73 \( 1 + (-0.258 - 0.965i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
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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

−18.217530659927577517873646022887, −17.68844190505052966213542128647, −17.01890812101724174766915347785, −16.22998784652070383460208583249, −15.935905885155908741902340505023, −15.00821326409923770080214780957, −14.597135335632735007913713619016, −13.85339653372585990191105801539, −13.027329355235071574521613040008, −12.14923047541144794559941131733, −11.6125412712733576272366406891, −10.98471359699351562259484734111, −10.173626634669298072007830132760, −9.503828238439679544463518614421, −9.22432122948431244827609113048, −8.16537576195614235735493368067, −7.40445358268276627342766633750, −6.64793782293565080402485086539, −5.7701463108122523235408938344, −5.11969069770476925790001911011, −4.42607739599566314387902383479, −3.7936407515661319002585578303, −2.959223325123952250041879451853, −2.05172931018773085287627835954, −1.04086110012032184400242085107, 0.58728357359915641943468034977, 0.96298608973516239583945008549, 2.33786434008047797945513729807, 2.76948502525624859040245175805, 3.6597337692783585262287678711, 4.79519385818088621528800101305, 5.551176972673003464231868504019, 6.06901322327842059363372605567, 6.81094674319900956388859796967, 7.70674868556258465975503932174, 8.07687328316621794813382656282, 8.84013781979779668707055275237, 9.716412604059432540359613158753, 10.650512020111339876457137478738, 11.16686419824685692245053325700, 11.82927176953005877901876170757, 12.55970681352143842268141527028, 13.29373061833965705569785195457, 13.57384342813256241678019958247, 14.47584458564995891232174494534, 15.12706068162717495534519520911, 16.099994742151318633213682901442, 16.50302800238815692430380233222, 17.46880257174020782164036519416, 17.82337620927128638056923246649

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