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Properties

Label 1-4009-4009.1086-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.998 - 0.0474i$
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  + (−0.124 + 0.992i)2-s + (0.921 − 0.388i)3-s + (−0.969 − 0.246i)4-s + (0.542 − 0.840i)5-s + (0.270 + 0.962i)6-s + (−0.955 + 0.294i)7-s + (0.365 − 0.930i)8-s + (0.698 − 0.715i)9-s + (0.766 + 0.642i)10-s + (−0.733 + 0.680i)11-s + (−0.988 + 0.149i)12-s + (−0.980 + 0.198i)13-s + (−0.173 − 0.984i)14-s + (0.173 − 0.984i)15-s + (0.878 + 0.478i)16-s + (0.318 + 0.947i)17-s + ⋯
L(s)  = 1  + (−0.124 + 0.992i)2-s + (0.921 − 0.388i)3-s + (−0.969 − 0.246i)4-s + (0.542 − 0.840i)5-s + (0.270 + 0.962i)6-s + (−0.955 + 0.294i)7-s + (0.365 − 0.930i)8-s + (0.698 − 0.715i)9-s + (0.766 + 0.642i)10-s + (−0.733 + 0.680i)11-s + (−0.988 + 0.149i)12-s + (−0.980 + 0.198i)13-s + (−0.173 − 0.984i)14-s + (0.173 − 0.984i)15-s + (0.878 + 0.478i)16-s + (0.318 + 0.947i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0474i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0474i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.998 - 0.0474i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1086, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.998 - 0.0474i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.713045123 - 0.04064027346i\)
\(L(\frac12)\) \(\approx\) \(1.713045123 - 0.04064027346i\)
\(L(1)\) \(\approx\) \(1.130444879 + 0.2119010933i\)
\(L(1)\) \(\approx\) \(1.130444879 + 0.2119010933i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.124 + 0.992i)T \)
3 \( 1 + (0.921 - 0.388i)T \)
5 \( 1 + (0.542 - 0.840i)T \)
7 \( 1 + (-0.955 + 0.294i)T \)
11 \( 1 + (-0.733 + 0.680i)T \)
13 \( 1 + (-0.980 + 0.198i)T \)
17 \( 1 + (0.318 + 0.947i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.318 + 0.947i)T \)
31 \( 1 + (0.365 - 0.930i)T \)
37 \( 1 + (0.733 + 0.680i)T \)
41 \( 1 + (-0.318 - 0.947i)T \)
43 \( 1 + (-0.661 + 0.749i)T \)
47 \( 1 + (-0.411 - 0.911i)T \)
53 \( 1 + (0.583 - 0.811i)T \)
59 \( 1 + (0.318 + 0.947i)T \)
61 \( 1 + (-0.766 - 0.642i)T \)
67 \( 1 + (0.921 + 0.388i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
73 \( 1 + (0.995 - 0.0995i)T \)
79 \( 1 + (0.0249 + 0.999i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.456 - 0.889i)T \)
97 \( 1 + (-0.583 - 0.811i)T \)
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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

−18.65218021112342603166963403505, −18.18764184407452739517559559294, −17.15866004142343462284182112827, −16.560763008910437247745469218786, −15.64313044852652164304053068481, −14.938565646090339928587411285832, −14.14199821256304057063833383619, −13.65836926687016789292253728743, −13.15966117106711973461930726498, −12.458367420907302453352867075453, −11.40518530638913401726128251249, −10.68723879082428816362918915896, −10.16184027859021593233254196321, −9.52133575916082976307985350856, −9.233633476155078732773884987105, −8.06352071194017735559864738254, −7.50771741998125566782512872189, −6.68247912007385287898825477340, −5.49477261474926343512664982143, −4.873568432245423108222231996086, −3.814420200900961030200081092245, −3.011959954973500784377791503351, −2.85501326035366947949320566243, −2.0793879869915595987289519112, −0.80838240703485140694686877532, 0.5637342447009598041078831785, 1.68635582693811650172154561659, 2.48640843545419349426590513156, 3.45064149932732116262088398745, 4.379303931904918256124516337356, 5.09712045143257479662638034068, 5.836542705339852733021811026495, 6.73570676090960125154188383413, 7.176496124945025620250217449305, 8.16413598931712981833971678143, 8.53714724919879027411579422519, 9.50252552722833166893009655853, 9.69916018093940264062218337791, 10.39370423339537898481645977214, 12.09978169376104090814582617865, 12.76598295434573279495007403989, 13.00864194652618919543233440798, 13.64007772747698623089344621777, 14.58361544259729263412049285450, 15.072368317712936791330200848433, 15.614751444594241403974327768316, 16.58084965378085517874787099046, 16.92042228348378044977515834128, 17.74565328454527187418567846884, 18.51534521471622553951982253720

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