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Properties

Label 1-4009-4009.1023-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.935 - 0.352i$
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  + (−0.0249 − 0.999i)2-s + (−0.853 + 0.521i)3-s + (−0.998 + 0.0498i)4-s + (−0.318 − 0.947i)5-s + (0.542 + 0.840i)6-s + (0.623 + 0.781i)7-s + (0.0747 + 0.997i)8-s + (0.456 − 0.889i)9-s + (−0.939 + 0.342i)10-s + (−0.988 − 0.149i)11-s + (0.826 − 0.563i)12-s + (−0.969 − 0.246i)13-s + (0.766 − 0.642i)14-s + (0.766 + 0.642i)15-s + (0.995 − 0.0995i)16-s + (−0.969 + 0.246i)17-s + ⋯
L(s)  = 1  + (−0.0249 − 0.999i)2-s + (−0.853 + 0.521i)3-s + (−0.998 + 0.0498i)4-s + (−0.318 − 0.947i)5-s + (0.542 + 0.840i)6-s + (0.623 + 0.781i)7-s + (0.0747 + 0.997i)8-s + (0.456 − 0.889i)9-s + (−0.939 + 0.342i)10-s + (−0.988 − 0.149i)11-s + (0.826 − 0.563i)12-s + (−0.969 − 0.246i)13-s + (0.766 − 0.642i)14-s + (0.766 + 0.642i)15-s + (0.995 − 0.0995i)16-s + (−0.969 + 0.246i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.352i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.935 - 0.352i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1023, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.935 - 0.352i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2634848636 - 0.04795219683i\)
\(L(\frac12)\) \(\approx\) \(0.2634848636 - 0.04795219683i\)
\(L(1)\) \(\approx\) \(0.4457698941 - 0.2171870784i\)
\(L(1)\) \(\approx\) \(0.4457698941 - 0.2171870784i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.0249 - 0.999i)T \)
3 \( 1 + (-0.853 + 0.521i)T \)
5 \( 1 + (-0.318 - 0.947i)T \)
7 \( 1 + (0.623 + 0.781i)T \)
11 \( 1 + (-0.988 - 0.149i)T \)
13 \( 1 + (-0.969 - 0.246i)T \)
17 \( 1 + (-0.969 + 0.246i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (-0.969 - 0.246i)T \)
31 \( 1 + (-0.900 - 0.433i)T \)
37 \( 1 + (0.365 - 0.930i)T \)
41 \( 1 + (0.270 - 0.962i)T \)
43 \( 1 + (-0.969 + 0.246i)T \)
47 \( 1 + (-0.124 - 0.992i)T \)
53 \( 1 + (-0.797 + 0.603i)T \)
59 \( 1 + (-0.969 + 0.246i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.878 - 0.478i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (0.921 - 0.388i)T \)
79 \( 1 + (-0.411 + 0.911i)T \)
83 \( 1 + T \)
89 \( 1 + (0.980 - 0.198i)T \)
97 \( 1 + (0.921 - 0.388i)T \)
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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

−18.324435022052602069696284352269, −17.751045634236299836591394321, −17.31312250175280210956013052710, −16.41375676844779399699546038728, −16.00333304980913594415482713039, −15.03945553973890020800735732467, −14.58458634064561485015850557290, −13.7802091795219791953498588712, −13.22034657178262045283998120450, −12.48330863632310766618931853352, −11.54893596765708659019407463693, −10.93977298881489364758003924433, −10.262362899818430775436933254737, −9.66968348014725021766675434939, −8.35359358465788058652172644561, −7.668809686178360420125703559873, −7.34268070184233525250066410725, −6.66875748047163406843195513402, −6.03516591682938695325274965142, −4.97894427071481272370532239255, −4.67595529671221117357472777825, −3.735278403548940849109195188375, −2.522258639527433388189157622682, −1.59514378860673071178122003789, −0.19999553835939505175540163604, 0.40706072135403843122534809372, 1.77078044315787185258169692836, 2.262584233510468600062416539309, 3.5143048718167023985752788407, 4.23512421789413148577340955653, 4.992419504107962464264921433918, 5.32111282837480525893893586076, 6.0222250947280812703553922643, 7.56295371560937220017823030770, 8.083370594591094195821415839527, 9.07666544351034481428573638373, 9.39646516602542424085459011456, 10.30185555257618732089000734226, 11.01542254705347606168488479658, 11.53126753778081072134904979034, 12.20870212450533029517879851667, 12.70182951991442253232351251438, 13.26914278461007375410830025944, 14.351897223710193980518383519025, 15.25330578019368427507833358323, 15.61316211220305888515849457045, 16.606463751125606082555172775364, 17.16931440765619365113836843268, 17.888293553908034700017813124936, 18.29828514812556095932013385722

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