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Properties

Label 1-6001-6001.121-r0-0-0
Degree $1$
Conductor $6001$
Sign $0.737 + 0.675i$
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  + (0.540 + 0.841i)2-s + (0.212 + 0.977i)3-s + (−0.415 + 0.909i)4-s + (−0.212 − 0.977i)5-s + (−0.707 + 0.707i)6-s + (−0.707 + 0.707i)7-s + (−0.989 + 0.142i)8-s + (−0.909 + 0.415i)9-s + (0.707 − 0.707i)10-s + (−0.877 − 0.479i)11-s + (−0.977 − 0.212i)12-s + (−0.755 + 0.654i)13-s + (−0.977 − 0.212i)14-s + (0.909 − 0.415i)15-s + (−0.654 − 0.755i)16-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)2-s + (0.212 + 0.977i)3-s + (−0.415 + 0.909i)4-s + (−0.212 − 0.977i)5-s + (−0.707 + 0.707i)6-s + (−0.707 + 0.707i)7-s + (−0.989 + 0.142i)8-s + (−0.909 + 0.415i)9-s + (0.707 − 0.707i)10-s + (−0.877 − 0.479i)11-s + (−0.977 − 0.212i)12-s + (−0.755 + 0.654i)13-s + (−0.977 − 0.212i)14-s + (0.909 − 0.415i)15-s + (−0.654 − 0.755i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6001\)    =    \(17 \cdot 353\)
Sign: $0.737 + 0.675i$
Analytic conductor: \(27.8685\)
Root analytic conductor: \(27.8685\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6001} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6001,\ (0:\ ),\ 0.737 + 0.675i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4716278110 + 0.1833239458i\)
\(L(\frac12)\) \(\approx\) \(0.4716278110 + 0.1833239458i\)
\(L(1)\) \(\approx\) \(0.6082252967 + 0.6246052399i\)
\(L(1)\) \(\approx\) \(0.6082252967 + 0.6246052399i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
353 \( 1 \)
good2 \( 1 + (0.540 + 0.841i)T \)
3 \( 1 + (0.212 + 0.977i)T \)
5 \( 1 + (-0.212 - 0.977i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (-0.877 - 0.479i)T \)
13 \( 1 + (-0.755 + 0.654i)T \)
19 \( 1 + (0.755 + 0.654i)T \)
23 \( 1 + (0.479 + 0.877i)T \)
29 \( 1 + (-0.936 - 0.349i)T \)
31 \( 1 + (-0.877 - 0.479i)T \)
37 \( 1 + (-0.599 + 0.800i)T \)
41 \( 1 + (-0.877 - 0.479i)T \)
43 \( 1 + (-0.281 - 0.959i)T \)
47 \( 1 + (0.415 - 0.909i)T \)
53 \( 1 + (0.841 + 0.540i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.0713 + 0.997i)T \)
67 \( 1 + iT \)
71 \( 1 + (-0.936 + 0.349i)T \)
73 \( 1 + (0.349 + 0.936i)T \)
79 \( 1 + (0.212 - 0.977i)T \)
83 \( 1 + (-0.989 - 0.142i)T \)
89 \( 1 + (0.989 + 0.142i)T \)
97 \( 1 + (-0.599 - 0.800i)T \)
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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

−17.97157367675962467457185801820, −17.29090906586089350054422010979, −16.25637817895593182825591431967, −15.31942479227368653394729881042, −14.84973805120382807992882574242, −14.186797149832716279397223543848, −13.594268068048676557277074680206, −12.91547806064594838723644432736, −12.57564538019947913596404879150, −11.8385431458646003616034037855, −10.82192312244379608354928424451, −10.74455805973733037848709764409, −9.7667118069880244850833425322, −9.2453117353185658744462114146, −8.09472641295771976916135154377, −7.314396107666704802776912160679, −6.98468750979959327348489990683, −6.1746426250057765285822594181, −5.37057939821123715691161611233, −4.627354565757707857217813102111, −3.44226074600777822599849881265, −3.12308420304007401789316854530, −2.487295798907398427304946122791, −1.73013449465568706452691261004, −0.57792193318605943376844962723, 0.15681451500556086341075501381, 1.95241050510300043039769418439, 2.88444537748890811245230597175, 3.565609191803848518166895393632, 4.13028411174632807395415279992, 5.1048470628982238570751450698, 5.44595474464739551357759492376, 5.87208293898034697235669676685, 7.14853076789201786428209414031, 7.698785382405718402839178774305, 8.63484183035966487943628607114, 8.92541671734581818421933165925, 9.61912922231151722578524836403, 10.23258543591083813152144989935, 11.55639318305158868965881911103, 11.859009529979080703198998389790, 12.64203292636415245914408234096, 13.48543128678264412910847213634, 13.70673196142347372696998284380, 14.85118597912305380550615506939, 15.245814392036193429130043192827, 15.85258332450986541279328240152, 16.32726465965526793504191086243, 16.8613262707219519384513565874, 17.30946969625765282550600858187

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