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Properties

Label 1-4009-4009.1075-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.386 - 0.922i$
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  + (−0.337 + 0.941i)2-s + (0.791 + 0.611i)3-s + (−0.772 − 0.635i)4-s + (0.791 − 0.611i)5-s + (−0.842 + 0.538i)6-s + (−0.337 − 0.941i)7-s + (0.858 − 0.512i)8-s + (0.251 + 0.967i)9-s + (0.309 + 0.951i)10-s + (−0.691 + 0.722i)11-s + (−0.222 − 0.974i)12-s + (0.193 + 0.981i)13-s + 14-s + 15-s + (0.193 + 0.981i)16-s + (−0.599 − 0.800i)17-s + ⋯
L(s)  = 1  + (−0.337 + 0.941i)2-s + (0.791 + 0.611i)3-s + (−0.772 − 0.635i)4-s + (0.791 − 0.611i)5-s + (−0.842 + 0.538i)6-s + (−0.337 − 0.941i)7-s + (0.858 − 0.512i)8-s + (0.251 + 0.967i)9-s + (0.309 + 0.951i)10-s + (−0.691 + 0.722i)11-s + (−0.222 − 0.974i)12-s + (0.193 + 0.981i)13-s + 14-s + 15-s + (0.193 + 0.981i)16-s + (−0.599 − 0.800i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1075, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5931796150 - 0.3947514575i\)
\(L(\frac12)\) \(\approx\) \(0.5931796150 - 0.3947514575i\)
\(L(1)\) \(\approx\) \(0.8931695640 + 0.3415164006i\)
\(L(1)\) \(\approx\) \(0.8931695640 + 0.3415164006i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (-0.337 + 0.941i)T \)
3 \( 1 + (0.791 + 0.611i)T \)
5 \( 1 + (0.791 - 0.611i)T \)
7 \( 1 + (-0.337 - 0.941i)T \)
11 \( 1 + (-0.691 + 0.722i)T \)
13 \( 1 + (0.193 + 0.981i)T \)
17 \( 1 + (-0.599 - 0.800i)T \)
23 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (-0.946 - 0.323i)T \)
31 \( 1 + (-0.733 + 0.680i)T \)
37 \( 1 + (-0.280 + 0.959i)T \)
41 \( 1 + (-0.995 - 0.0896i)T \)
43 \( 1 + (-0.222 - 0.974i)T \)
47 \( 1 + (-0.963 + 0.266i)T \)
53 \( 1 + (0.936 + 0.351i)T \)
59 \( 1 + (0.575 - 0.817i)T \)
61 \( 1 + (0.669 - 0.743i)T \)
67 \( 1 + (0.0747 - 0.997i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.623 + 0.781i)T \)
79 \( 1 + (0.0149 - 0.999i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (-0.999 - 0.0299i)T \)
97 \( 1 + (0.998 + 0.0598i)T \)
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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

−18.49363586964870551447629185600, −18.21083545377201652905045327655, −17.72979599279155818997044728105, −16.82227385817026997456371738428, −15.764730642171861114384889001269, −15.00135304576812627084311609069, −14.45923644210617959911084674090, −13.37735870583329007691127190233, −13.172812267368145617635087904184, −12.7264184433716118482024148936, −11.645734580696746573500701382498, −11.058351560764372317698868822370, −10.20784414916715106171624048793, −9.643391434273160149101738997730, −8.89459658922987608292817760871, −8.3775106033196179795295558407, −7.65476276958666340472490041853, −6.78693291097091565030124358880, −5.72809944155926259837000549184, −5.41119342734341124355901288406, −3.75321541700473152078972055414, −3.32816832092774239840465009159, −2.520559553521156284336356822435, −2.09947814362873569652508587593, −1.227859326386296458818790020222, 0.185413861800327002607720993552, 1.58753340095345844657401258020, 2.19162634012429902325990226653, 3.466136358525296171796870798094, 4.38284183666190291516380276661, 4.79732875016659039612438992302, 5.48628285722188800124047113301, 6.74975800135444174258132607754, 6.98034742916431564823065110016, 8.01426818014731435901656606239, 8.63085848399055008666196289409, 9.33303211579308349484638603474, 9.83253873636974499620200873124, 10.33683242919666614277236652394, 11.13710970356897690782970057196, 12.59337073882593581586478947187, 13.23362168525620162764633686945, 13.809220523935601238928893617003, 14.16321463483310572550990206118, 15.05417822817736231157261195132, 15.712974899465425777442967341513, 16.48379135555517625515449156573, 16.679780148013596332795458745857, 17.48693793449223372964011971856, 18.337097226835513848000545237795

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