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Properties

Label 1-4009-4009.1039-r0-0-0
Degree $1$
Conductor $4009$
Sign $-0.945 - 0.325i$
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  + (0.289 − 0.957i)2-s + (0.905 − 0.424i)3-s + (−0.831 − 0.555i)4-s + (0.966 − 0.256i)5-s + (−0.144 − 0.989i)6-s + (0.393 − 0.919i)7-s + (−0.772 + 0.635i)8-s + (0.638 − 0.769i)9-s + (0.0348 − 0.999i)10-s + (−0.873 + 0.486i)11-s + (−0.988 − 0.149i)12-s + (0.676 + 0.736i)13-s + (−0.766 − 0.642i)14-s + (0.766 − 0.642i)15-s + (0.383 + 0.923i)16-s + (0.909 + 0.415i)17-s + ⋯
L(s)  = 1  + (0.289 − 0.957i)2-s + (0.905 − 0.424i)3-s + (−0.831 − 0.555i)4-s + (0.966 − 0.256i)5-s + (−0.144 − 0.989i)6-s + (0.393 − 0.919i)7-s + (−0.772 + 0.635i)8-s + (0.638 − 0.769i)9-s + (0.0348 − 0.999i)10-s + (−0.873 + 0.486i)11-s + (−0.988 − 0.149i)12-s + (0.676 + 0.736i)13-s + (−0.766 − 0.642i)14-s + (0.766 − 0.642i)15-s + (0.383 + 0.923i)16-s + (0.909 + 0.415i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.945 - 0.325i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $-0.945 - 0.325i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ -0.945 - 0.325i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5474665960 - 3.270505683i\)
\(L(\frac12)\) \(\approx\) \(0.5474665960 - 3.270505683i\)
\(L(1)\) \(\approx\) \(1.233506446 - 1.379593569i\)
\(L(1)\) \(\approx\) \(1.233506446 - 1.379593569i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.289 - 0.957i)T \)
3 \( 1 + (0.905 - 0.424i)T \)
5 \( 1 + (0.966 - 0.256i)T \)
7 \( 1 + (0.393 - 0.919i)T \)
11 \( 1 + (-0.873 + 0.486i)T \)
13 \( 1 + (0.676 + 0.736i)T \)
17 \( 1 + (0.909 + 0.415i)T \)
23 \( 1 + (-0.961 + 0.275i)T \)
29 \( 1 + (-0.676 - 0.736i)T \)
31 \( 1 + (0.623 - 0.781i)T \)
37 \( 1 + (-0.0149 - 0.999i)T \)
41 \( 1 + (-0.917 + 0.397i)T \)
43 \( 1 + (0.980 - 0.198i)T \)
47 \( 1 + (-0.863 - 0.504i)T \)
53 \( 1 + (0.993 + 0.109i)T \)
59 \( 1 + (-0.114 + 0.993i)T \)
61 \( 1 + (0.882 - 0.469i)T \)
67 \( 1 + (0.921 - 0.388i)T \)
71 \( 1 + (0.615 + 0.788i)T \)
73 \( 1 + (-0.583 + 0.811i)T \)
79 \( 1 + (-0.943 - 0.332i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (0.153 - 0.988i)T \)
97 \( 1 + (-0.952 - 0.304i)T \)
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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

−18.49134775109296022472246637059, −18.206100547686256554899820236427, −17.47051579912313708363229249506, −16.40839079938340433373612390620, −16.015716095692109045497229006795, −15.29074295308368291151011583619, −14.741836758736246793128295245830, −14.064755590922673691417417909724, −13.63111897198156534623760681376, −12.91612100987114156127553937109, −12.2747680229982705360919383384, −11.06156330203971909383245795705, −10.193119965762566519127262627913, −9.694797571839108661848743389973, −8.833173529566678974591086537093, −8.297285869334193544385334806645, −7.8373139406094684229160604227, −6.82462841306580706583057065414, −5.9401960075909531449980469301, −5.324159296890014206159559838436, −4.93742148909330176109280730728, −3.65866796250621934572676232680, −3.02861696954199965520553974370, −2.41586837560274851322120411408, −1.27464952864101466877053854171, 0.72999054644164433488488937552, 1.646503922528526574114574183697, 2.013881209598477314513380421280, 2.86354734991736887450335452915, 3.91420267172444133654538242285, 4.24019492769902024175009900710, 5.35469292492497738105736984978, 6.02025431087060363259892930314, 6.9965073780723656392221520815, 7.92992827113259872506635285002, 8.4475869593528859484080864014, 9.382933711042840397641311280509, 9.97188277363846636703250072565, 10.34247859102864708859860854434, 11.34149769583296202406425372209, 12.11987711164879202921008134789, 13.00167675416325921587300205967, 13.30096534258768900195289001180, 13.95003211968475705278310260705, 14.382799506742598835558095449195, 15.11323793669187667137440076971, 16.12032028017411128868795731490, 17.11389082303827737518571429344, 17.70029864539686257727454891432, 18.395533175780164989505408533270

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