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Properties

Label 1-4009-4009.1071-r0-0-0
Degree $1$
Conductor $4009$
Sign $-0.718 + 0.695i$
Analytic cond. $18.6177$
Root an. cond. $18.6177$
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  + (−0.599 + 0.800i)2-s + (0.575 + 0.817i)3-s + (−0.280 − 0.959i)4-s + (0.575 − 0.817i)5-s + (−0.999 − 0.0299i)6-s + (−0.599 − 0.800i)7-s + (0.936 + 0.351i)8-s + (−0.337 + 0.941i)9-s + (0.309 + 0.951i)10-s + (0.858 + 0.512i)11-s + (0.623 − 0.781i)12-s + (−0.842 + 0.538i)13-s + 14-s + 15-s + (−0.842 + 0.538i)16-s + (0.712 + 0.701i)17-s + ⋯
L(s)  = 1  + (−0.599 + 0.800i)2-s + (0.575 + 0.817i)3-s + (−0.280 − 0.959i)4-s + (0.575 − 0.817i)5-s + (−0.999 − 0.0299i)6-s + (−0.599 − 0.800i)7-s + (0.936 + 0.351i)8-s + (−0.337 + 0.941i)9-s + (0.309 + 0.951i)10-s + (0.858 + 0.512i)11-s + (0.623 − 0.781i)12-s + (−0.842 + 0.538i)13-s + 14-s + 15-s + (−0.842 + 0.538i)16-s + (0.712 + 0.701i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $-0.718 + 0.695i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1071, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ -0.718 + 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5357987979 + 1.323765124i\)
\(L(\frac12)\) \(\approx\) \(0.5357987979 + 1.323765124i\)
\(L(1)\) \(\approx\) \(0.8357051938 + 0.5239627932i\)
\(L(1)\) \(\approx\) \(0.8357051938 + 0.5239627932i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.599 + 0.800i)T \)
3 \( 1 + (0.575 + 0.817i)T \)
5 \( 1 + (0.575 - 0.817i)T \)
7 \( 1 + (-0.599 - 0.800i)T \)
11 \( 1 + (0.858 + 0.512i)T \)
13 \( 1 + (-0.842 + 0.538i)T \)
17 \( 1 + (0.712 + 0.701i)T \)
23 \( 1 + (0.913 + 0.406i)T \)
29 \( 1 + (0.887 + 0.460i)T \)
31 \( 1 + (0.365 + 0.930i)T \)
37 \( 1 + (0.0149 + 0.999i)T \)
41 \( 1 + (-0.550 - 0.834i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (0.983 + 0.178i)T \)
53 \( 1 + (-0.691 - 0.722i)T \)
59 \( 1 + (-0.447 + 0.894i)T \)
61 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + (-0.733 + 0.680i)T \)
71 \( 1 + (-0.978 - 0.207i)T \)
73 \( 1 + (-0.900 - 0.433i)T \)
79 \( 1 + (-0.163 + 0.986i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (-0.946 - 0.323i)T \)
97 \( 1 + (0.791 + 0.611i)T \)
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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

−18.5198797280203008172739491565, −17.70667552447740229357221373520, −17.22158994031886927863549497517, −16.4293848393483107620953605304, −15.36978071427159047595659416666, −14.63926973574473715160297403008, −14.01160824898623120186113064318, −13.34219539491404815829830411117, −12.63773289120634163089986229805, −12.03427976782004078020015256188, −11.48672262095365826350684193065, −10.56514266380238656368729813148, −9.67547076164121719478314916454, −9.35847169454563848843321930918, −8.660347706623863832959182783565, −7.73748362436382165986308111384, −7.19560666997321030278737757049, −6.36315384857853504171160641242, −5.767099627001511950638005453102, −4.477470959715294033062048244621, −3.18667648145932769404712887540, −2.98941462244077940613649444351, −2.3711801538900559762488370934, −1.451735014421151390344157672295, −0.48823785938286381880760373706, 1.09818870821039361526403352956, 1.730108245939082523326878838985, 2.921170439300180737020459496011, 4.00241009919310877166204612513, 4.61021541322610115019170255881, 5.20319455518524974195710970135, 6.124282487365505652181473155371, 6.94636770980287516380343284930, 7.54662917118670351843195058819, 8.54107500745682380712240517396, 9.03992366948223647218489730045, 9.606501407864080935800499414234, 10.20373863370950865878210521324, 10.61525174719786683749182168673, 11.92208421974762366221537732952, 12.70834898412537641717870292120, 13.74180942777830066920432639625, 13.98485038636388748518758232255, 14.74855964780755648834750033131, 15.40337176728501972063369152332, 16.217314048522254672465154337500, 16.71662070200694750790454527443, 17.20592539748423621162065840005, 17.56988918888374933630306858297, 19.009342586707837460235243508635

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