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Properties

Label 1-6023-6023.1584-r0-0-0
Degree $1$
Conductor $6023$
Sign $-0.977 + 0.211i$
Analytic cond. $27.9706$
Root an. cond. $27.9706$
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  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + 7-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + 11-s − 12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)4-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)6-s + 7-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 − 0.866i)10-s + 11-s − 12-s + (0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6023 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6023\)    =    \(19 \cdot 317\)
Sign: $-0.977 + 0.211i$
Analytic conductor: \(27.9706\)
Root analytic conductor: \(27.9706\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6023} (1584, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6023,\ (0:\ ),\ -0.977 + 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4183214807 - 3.915758222i\)
\(L(\frac12)\) \(\approx\) \(-0.4183214807 - 3.915758222i\)
\(L(1)\) \(\approx\) \(1.026515902 - 1.712337500i\)
\(L(1)\) \(\approx\) \(1.026515902 - 1.712337500i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
317 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 + (0.5 + 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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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

−17.73194168317832003466551272104, −17.31532012096232828587676953131, −16.8121941024952223556713715446, −15.832015105084850126551250483393, −15.234884683081066308363188851, −14.834447832950406635886464171130, −14.330925952491874794216075208941, −13.69459146230952664193593533182, −13.28841039068336796288739573890, −12.02944625025646066660992441865, −11.51982324447049183541367137447, −10.68292553813410364060012185314, −10.04989504805910234923486750515, −9.341646149187016998478550932932, −8.56097492000855704913552052300, −7.94582152144515955896589878447, −7.50209159383249654362432915013, −6.33151919826067887679376107051, −5.91560114608457964022324555733, −5.22383373802398417034438540377, −4.319515327603094638001604337908, −3.82239398051555765095945996478, −3.10854592392366413316391547659, −2.34864429574270319608093406176, −1.294778703966301095935509147210, 0.823659679359154165464829701675, 1.27874632542370388620259759537, 1.90227850506047171868547551201, 2.595405679676453024096971044874, 3.52628194180804371362897941196, 4.49089850425817516630820342551, 4.746619305110871619580134190706, 5.9353637914999679394018322939, 6.27719513251183551590169879672, 7.25206474642874286828018428800, 8.27940329862581513266160201972, 8.72364910834970116259573757932, 9.338882740533386933252644357802, 9.9241788792579209043123998256, 11.08469240845540942465080699759, 11.56600780128821748460764113666, 12.30162412814496933886261150284, 12.50975581208365543816899553934, 13.57966731574387456739598296248, 14.04781701637736436224771972889, 14.24120354364216061357988649098, 15.02482504646768821255080891137, 16.08433204099797376654873285531, 16.86324618358776907158180200962, 17.61074101197779850229228810695

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