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Properties

Label 1-6001-6001.140-r0-0-0
Degree $1$
Conductor $6001$
Sign $-0.972 - 0.233i$
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  + (−0.841 + 0.540i)2-s + (0.540 − 0.841i)3-s + (0.415 − 0.909i)4-s + (0.540 − 0.841i)5-s + i·6-s + i·7-s + (0.142 + 0.989i)8-s + (−0.415 − 0.909i)9-s + i·10-s + (−0.281 − 0.959i)11-s + (−0.540 − 0.841i)12-s + (−0.654 − 0.755i)13-s + (−0.540 − 0.841i)14-s + (−0.415 − 0.909i)15-s + (−0.654 − 0.755i)16-s + ⋯
L(s)  = 1  + (−0.841 + 0.540i)2-s + (0.540 − 0.841i)3-s + (0.415 − 0.909i)4-s + (0.540 − 0.841i)5-s + i·6-s + i·7-s + (0.142 + 0.989i)8-s + (−0.415 − 0.909i)9-s + i·10-s + (−0.281 − 0.959i)11-s + (−0.540 − 0.841i)12-s + (−0.654 − 0.755i)13-s + (−0.540 − 0.841i)14-s + (−0.415 − 0.909i)15-s + (−0.654 − 0.755i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $-0.972 - 0.233i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ -0.972 - 0.233i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1413565101 - 1.195034162i\)
\(L(\frac12)\) \(\approx\) \(0.1413565101 - 1.195034162i\)
\(L(1)\) \(\approx\) \(0.7940081768 - 0.3808013355i\)
\(L(1)\) \(\approx\) \(0.7940081768 - 0.3808013355i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (-0.841 + 0.540i)T \)
3 \( 1 + (0.540 - 0.841i)T \)
5 \( 1 + (0.540 - 0.841i)T \)
7 \( 1 + iT \)
11 \( 1 + (-0.281 - 0.959i)T \)
13 \( 1 + (-0.654 - 0.755i)T \)
19 \( 1 + (0.654 - 0.755i)T \)
23 \( 1 + (0.281 - 0.959i)T \)
29 \( 1 + (0.909 - 0.415i)T \)
31 \( 1 + (0.281 + 0.959i)T \)
37 \( 1 + (-0.989 + 0.142i)T \)
41 \( 1 + (-0.281 - 0.959i)T \)
43 \( 1 + (0.959 - 0.281i)T \)
47 \( 1 + (0.415 - 0.909i)T \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 - T \)
61 \( 1 + (0.755 - 0.654i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.909 - 0.415i)T \)
73 \( 1 + (-0.909 - 0.415i)T \)
79 \( 1 + (0.540 + 0.841i)T \)
83 \( 1 + (0.142 - 0.989i)T \)
89 \( 1 + (-0.142 + 0.989i)T \)
97 \( 1 + (0.989 + 0.142i)T \)
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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

−17.94957101017922024732198792211, −17.28342126556516722158699637137, −16.99807034850021447601420022201, −16.06048352075650233677443234414, −15.578096571862618955766104801968, −14.66874673244155744962651710730, −14.14648758351235176649387786265, −13.5477832002060451297825974185, −12.753062526711489860612547863261, −11.80215042880275541065890213645, −11.158132178623610239200988715557, −10.50622619101440692389696494954, −9.9149355692637994379978066152, −9.6961133097625234244966752456, −8.9909125539587853675343914437, −7.879022070201476505661212416426, −7.46243116530622899815037849731, −6.927696605261478563174296287664, −5.92470383600351390361536320680, −4.76359737584576625836406403425, −4.20214227271945304416595958479, −3.35822870567147369316980371501, −2.81829529474645484429059266843, −1.99673830878797713562384717856, −1.34672178572601671332732328838, 0.39840531124720422553089171805, 0.98133283441842371754065835260, 2.00890812956470531440223353794, 2.548157230996327944243732493752, 3.24684971928676778282620780751, 4.92190279561966756243764587026, 5.29298539168884254448229410377, 6.07260891682834970109404077095, 6.609036418928190679136206605900, 7.503430383379392782548775324265, 8.20149545156033911880126401277, 8.78713953495004819747067057168, 8.96943283225299464761349693522, 9.8942245120671392651733565414, 10.5665817581447379093506255161, 11.58617539431920301523693627963, 12.23337765242758299684553517511, 12.75270889593282064444290558442, 13.686727167517675839343187942674, 14.07232634711820248608381291288, 14.846072652308964958430072066779, 15.755764867996491935310232881670, 15.89020881858858628579064494285, 17.01868821434937375232706301919, 17.50448711830521073969508076539

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