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Properties

Label 1-6001-6001.1058-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.615 - 0.788i$
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 + i·3-s + 4-s + i·5-s − i·6-s − i·7-s − 8-s − 9-s − i·10-s + i·11-s + i·12-s − 13-s + i·14-s − 15-s + 16-s + ⋯
L(s)  = 1  − 2-s + i·3-s + 4-s + i·5-s − i·6-s − i·7-s − 8-s − 9-s − i·10-s + i·11-s + i·12-s − 13-s + i·14-s − 15-s + 16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.615 - 0.788i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (1058, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.615 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004969278221 + 0.002424651797i\)
\(L(\frac12)\) \(\approx\) \(0.004969278221 + 0.002424651797i\)
\(L(1)\) \(\approx\) \(0.4443701037 + 0.2745825520i\)
\(L(1)\) \(\approx\) \(0.4443701037 + 0.2745825520i\)

Euler product

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

−17.13550123590488865041328239532, −16.70335494169986278995549840009, −16.175230538045625319906671710155, −15.23417218154775778260072531578, −14.68171952476089875960302515829, −13.84599091381162024745680021107, −12.83396933115924124492041950936, −12.49148184888644209820530788149, −11.96174591901351648412271370519, −11.2802423689788834221926887295, −10.56543931954814110923220542446, −9.55399698120823338152180229047, −8.88526742129481363860140241722, −8.510922906444681150940957867354, −8.02603468994699514086351898462, −7.13261710654427027502407915850, −6.47488248299654546401441750377, −5.634773882713865555090680042733, −5.354549684384231002998270947878, −4.03008983909799321681543959202, −2.844524019346740090132718675202, −2.30296089463028736146958636133, −1.68147994105189608556892723917, −0.66454950522633310644312634721, −0.00268789566549544176184965704, 1.45299059561552791750777892099, 2.389643178503963078432190909007, 2.97023624308492261822175326013, 3.849693360057323615984620108807, 4.492474807156420902163809984695, 5.43816939479021504910340010334, 6.39077026241229343223553348345, 6.98455120055003450821083800028, 7.559148784154024042824562713375, 8.23775267433713907964557387757, 9.17734373088692549748202617735, 9.94646801786535446905847821287, 10.17859675409002549543470407647, 10.62090302916639797802059698120, 11.594092439692301262090633093268, 11.82680573259201257841263972812, 13.02975448128549203157404593391, 13.93794296893288195620126670112, 14.75103943716011359152908976385, 15.051137005764624419174572095985, 15.59712319513258568449134026961, 16.51725297394717399675297586384, 17.059068729960668863104389284507, 17.53167672344202386731057163713, 18.01642533494357393231820134664

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