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Properties

Label 1-4009-4009.1025-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.606 - 0.794i$
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  + (0.525 − 0.850i)2-s + (0.971 + 0.237i)3-s + (−0.447 − 0.894i)4-s + (−0.691 − 0.722i)5-s + (0.712 − 0.701i)6-s + (0.999 − 0.0299i)7-s + (−0.995 − 0.0896i)8-s + (0.887 + 0.460i)9-s + (−0.978 + 0.207i)10-s + (0.134 − 0.990i)11-s + (−0.222 − 0.974i)12-s + (0.393 + 0.919i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.599 + 0.800i)16-s + (−0.193 + 0.981i)17-s + ⋯
L(s)  = 1  + (0.525 − 0.850i)2-s + (0.971 + 0.237i)3-s + (−0.447 − 0.894i)4-s + (−0.691 − 0.722i)5-s + (0.712 − 0.701i)6-s + (0.999 − 0.0299i)7-s + (−0.995 − 0.0896i)8-s + (0.887 + 0.460i)9-s + (−0.978 + 0.207i)10-s + (0.134 − 0.990i)11-s + (−0.222 − 0.974i)12-s + (0.393 + 0.919i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.599 + 0.800i)16-s + (−0.193 + 0.981i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.606 - 0.794i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (1025, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.606 - 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.054352228 - 1.510661186i\)
\(L(\frac12)\) \(\approx\) \(3.054352228 - 1.510661186i\)
\(L(1)\) \(\approx\) \(1.711749163 - 0.8103908329i\)
\(L(1)\) \(\approx\) \(1.711749163 - 0.8103908329i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.525 - 0.850i)T \)
3 \( 1 + (0.971 + 0.237i)T \)
5 \( 1 + (-0.691 - 0.722i)T \)
7 \( 1 + (0.999 - 0.0299i)T \)
11 \( 1 + (0.134 - 0.990i)T \)
13 \( 1 + (0.393 + 0.919i)T \)
17 \( 1 + (-0.193 + 0.981i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (0.992 + 0.119i)T \)
31 \( 1 + (-0.733 + 0.680i)T \)
37 \( 1 + (-0.791 + 0.611i)T \)
41 \( 1 + (0.0149 + 0.999i)T \)
43 \( 1 + (0.955 + 0.294i)T \)
47 \( 1 + (0.887 - 0.460i)T \)
53 \( 1 + (0.447 - 0.894i)T \)
59 \( 1 + (0.873 + 0.486i)T \)
61 \( 1 + (-0.669 - 0.743i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (-0.988 + 0.149i)T \)
79 \( 1 + (0.995 - 0.0896i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (0.983 - 0.178i)T \)
97 \( 1 + (0.936 - 0.351i)T \)
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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

−18.40571297163514287376847229894, −17.89044754513693447310185832112, −17.42237503459926747722539931094, −16.21156202137611478878803128218, −15.52705544341101554410136871647, −15.10438161302917344994915815202, −14.62828640312901806355031691014, −13.971058261864278017816155897860, −13.43000849819459216230166794463, −12.30719937991674502631103324522, −12.18776914940026441515715512387, −11.01544170128052971824731814577, −10.33766744222410766129036226038, −9.153860764606772335703796581674, −8.698737258776832218345630119616, −7.81377975450133574913430368813, −7.4004348121727471471685442679, −6.99362807228430307541697784264, −5.96839476195570707997813425199, −4.93599627313006282829227152027, −4.33209619350878459036745717330, −3.651632468468740611299062196611, −2.76805183306985731384637097227, −2.23016383627886983951416681605, −0.77897100128956647322455317689, 1.15519169001438161048162491912, 1.484568464992538524840859296899, 2.502716317257462057350192484489, 3.520217728579992917847959295982, 3.88726439423221071527584278599, 4.68817203151087634729993923602, 5.197366273672544462326579996654, 6.281482468775846990650379964220, 7.33698431425269782818551374280, 8.26093626591801970986392547683, 8.83668549060547730253991074078, 9.0459819558724232096075978578, 10.288196524934704351984959390864, 10.884742125689696530597626515808, 11.55962772495776654758659560426, 12.12377195449118778249547272086, 13.1150828893548349259563107831, 13.470341980217968363299301698018, 14.3334111319536061289598401482, 14.69559818744247673100053609413, 15.53032696622982937960990583771, 16.10039736736157249772154751617, 16.97617019800014533416186123588, 17.90047991723281662359637494519, 18.80712167296936567592219507525

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