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Properties

Label 1-6001-6001.2817-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.841 - 0.540i$
Analytic cond. $27.8685$
Root an. cond. $27.8685$
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  − 2-s + (−0.831 − 0.555i)3-s + 4-s + (−0.980 + 0.195i)5-s + (0.831 + 0.555i)6-s + (−0.195 + 0.980i)7-s − 8-s + (0.382 + 0.923i)9-s + (0.980 − 0.195i)10-s + (−0.923 − 0.382i)11-s + (−0.831 − 0.555i)12-s + (−0.980 − 0.195i)13-s + (0.195 − 0.980i)14-s + (0.923 + 0.382i)15-s + 16-s + ⋯
L(s)  = 1  − 2-s + (−0.831 − 0.555i)3-s + 4-s + (−0.980 + 0.195i)5-s + (0.831 + 0.555i)6-s + (−0.195 + 0.980i)7-s − 8-s + (0.382 + 0.923i)9-s + (0.980 − 0.195i)10-s + (−0.923 − 0.382i)11-s + (−0.831 − 0.555i)12-s + (−0.980 − 0.195i)13-s + (0.195 − 0.980i)14-s + (0.923 + 0.382i)15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.841 - 0.540i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.841 - 0.540i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (2817, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.841 - 0.540i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01290488329 - 0.04399959886i\)
\(L(\frac12)\) \(\approx\) \(0.01290488329 - 0.04399959886i\)
\(L(1)\) \(\approx\) \(0.3657597787 + 0.001568816274i\)
\(L(1)\) \(\approx\) \(0.3657597787 + 0.001568816274i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 - T \)
3 \( 1 + (-0.831 - 0.555i)T \)
5 \( 1 + (-0.980 + 0.195i)T \)
7 \( 1 + (-0.195 + 0.980i)T \)
11 \( 1 + (-0.923 - 0.382i)T \)
13 \( 1 + (-0.980 - 0.195i)T \)
19 \( 1 + (0.923 + 0.382i)T \)
23 \( 1 + (0.707 - 0.707i)T \)
29 \( 1 + (-0.923 + 0.382i)T \)
31 \( 1 + (0.980 - 0.195i)T \)
37 \( 1 + (-0.831 + 0.555i)T \)
41 \( 1 + (0.707 - 0.707i)T \)
43 \( 1 + (-0.382 + 0.923i)T \)
47 \( 1 + (-0.923 - 0.382i)T \)
53 \( 1 + (-0.195 - 0.980i)T \)
59 \( 1 + (0.195 + 0.980i)T \)
61 \( 1 + (0.382 - 0.923i)T \)
67 \( 1 + (-0.195 + 0.980i)T \)
71 \( 1 + (-0.980 - 0.195i)T \)
73 \( 1 + (-0.382 - 0.923i)T \)
79 \( 1 + (-0.980 - 0.195i)T \)
83 \( 1 + iT \)
89 \( 1 + (-0.195 + 0.980i)T \)
97 \( 1 + (0.923 + 0.382i)T \)
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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

−17.75381660376846201018053813094, −17.2868549546583125149123972636, −16.76913942719555751067850656679, −16.06594353614235181146330364222, −15.65686051162882166132659110282, −15.12375956349528929499824244235, −14.30509422137969512876346629166, −13.080652752647824282349131321122, −12.54433373987213055176235365703, −11.65138500175369771549800790899, −11.40041826133300780219791681886, −10.602774846375072693980213861818, −10.05081990740112607436171867056, −9.527765145582888282787624969726, −8.751679631164547683358120822654, −7.6849543312591388964974574346, −7.34915989276803954354753217369, −6.88996064984489595297935288970, −5.8138788514163356885022806298, −4.9919093243287770154786443178, −4.442268862083969982091482658120, −3.45263527359093865305393828849, −2.893709897923941480696415935835, −1.57442698005862265066295332921, −0.666366454220979393705267722776, 0.03454153246646514064340372986, 0.914162353165592780276234154695, 1.99777824243940924144920245183, 2.75835342469497084288765158854, 3.30131358681598622532800411999, 4.7640028238151602375837569976, 5.34347087265846214525894031001, 6.05552203739097975743331633389, 6.893998854939159705475565278912, 7.38115419203828140217690386505, 8.08269334506633423147272850687, 8.51223951459587687403607274996, 9.52685248720930290964668399433, 10.24607045959217121829649489957, 10.83345920166496929188209233822, 11.649786463823409688133238107270, 11.86118311307114874138172006737, 12.61921326751119113930459009379, 13.124160525280160958691577659196, 14.45829181232066856848087356879, 15.08219972466072963207018390438, 15.76724460762169323115847854932, 16.22239582366811634633644786746, 16.75208580714995082164202336553, 17.633700775054483092812062215076

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