↑

Properties

Label 1-4009-4009.108-r0-0-0
Degree $1$
Conductor $4009$
Sign $0.987 + 0.160i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.814 − 0.579i)2-s + (−0.917 + 0.397i)3-s + (0.327 − 0.944i)4-s + (0.232 − 0.972i)5-s + (−0.517 + 0.855i)6-s + (−0.712 + 0.701i)7-s + (−0.280 − 0.959i)8-s + (0.683 − 0.729i)9-s + (−0.374 − 0.927i)10-s + (−0.772 + 0.635i)11-s + (0.0747 + 0.997i)12-s + (0.203 + 0.979i)13-s + (−0.173 + 0.984i)14-s + (0.173 + 0.984i)15-s + (−0.784 − 0.619i)16-s + (0.863 − 0.504i)17-s + ⋯
L(s)  = 1  + (0.814 − 0.579i)2-s + (−0.917 + 0.397i)3-s + (0.327 − 0.944i)4-s + (0.232 − 0.972i)5-s + (−0.517 + 0.855i)6-s + (−0.712 + 0.701i)7-s + (−0.280 − 0.959i)8-s + (0.683 − 0.729i)9-s + (−0.374 − 0.927i)10-s + (−0.772 + 0.635i)11-s + (0.0747 + 0.997i)12-s + (0.203 + 0.979i)13-s + (−0.173 + 0.984i)14-s + (0.173 + 0.984i)15-s + (−0.784 − 0.619i)16-s + (0.863 − 0.504i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4009\)    =    \(19 \cdot 211\)
Sign: $0.987 + 0.160i$
Analytic conductor: \(18.6177\)
Root analytic conductor: \(18.6177\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4009} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4009,\ (0:\ ),\ 0.987 + 0.160i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.311795085 + 0.1060799718i\)
\(L(\frac12)\) \(\approx\) \(1.311795085 + 0.1060799718i\)
\(L(1)\) \(\approx\) \(1.036960067 - 0.3044782497i\)
\(L(1)\) \(\approx\) \(1.036960067 - 0.3044782497i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad19 \( 1 \)
211 \( 1 \)
good2 \( 1 + (0.814 - 0.579i)T \)
3 \( 1 + (-0.917 + 0.397i)T \)
5 \( 1 + (0.232 - 0.972i)T \)
7 \( 1 + (-0.712 + 0.701i)T \)
11 \( 1 + (-0.772 + 0.635i)T \)
13 \( 1 + (0.203 + 0.979i)T \)
17 \( 1 + (0.863 - 0.504i)T \)
23 \( 1 + (-0.559 - 0.829i)T \)
29 \( 1 + (-0.746 + 0.665i)T \)
31 \( 1 + (0.826 - 0.563i)T \)
37 \( 1 + (0.772 + 0.635i)T \)
41 \( 1 + (0.212 + 0.977i)T \)
43 \( 1 + (-0.583 + 0.811i)T \)
47 \( 1 + (-0.892 - 0.451i)T \)
53 \( 1 + (0.356 - 0.934i)T \)
59 \( 1 + (-0.212 - 0.977i)T \)
61 \( 1 + (0.374 + 0.927i)T \)
67 \( 1 + (-0.661 + 0.749i)T \)
71 \( 1 + (-0.0348 - 0.999i)T \)
73 \( 1 + (0.542 + 0.840i)T \)
79 \( 1 + (0.0647 + 0.997i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (0.607 + 0.794i)T \)
97 \( 1 + (-0.261 + 0.965i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.252193321137080378476661249997, −17.6634212570330848551881426266, −17.03498308321382925302950808166, −16.41919059820820426689234040461, −15.66089620056889257021877108890, −15.27085974154052630765480083206, −14.156138984380802762952425520668, −13.64103201623794018882640023006, −13.109794626025282940497929171413, −12.48848849283051650596959273599, −11.66992468736783047612893786697, −10.89554404604165338494508484613, −10.44085666720613069285073262058, −9.75216878599492057532247303948, −8.27340974757641486645000000704, −7.54251945868782465112318811596, −7.26677629096276665874145696627, −6.18463988777411318758697494688, −5.926543428204875767364987014166, −5.33762170784223756135276888055, −4.19511775576985236667360338745, −3.41478477646374280521499540388, −2.863644318606893739080325023989, −1.77070630972713055403368997582, −0.382163422239479501972949763896, 0.83046512790114162642973484776, 1.756767152171169042022628952857, 2.600925702473610922060804895587, 3.59143507073703150389468508997, 4.45904163938702102823734940608, 4.94364191775326012527596212266, 5.58033891731700694248466667314, 6.257258495913370372755215409592, 6.84983014965043627186129452620, 8.08605737139913326051339503254, 9.16788355706766391147966692974, 9.91123956485165301574975438797, 9.93782350535373876219815408714, 11.18544845767819066893138616824, 11.83617490276118243818909568370, 12.227528639885671936505655871147, 13.01238469512630255804603107904, 13.2744673073921360724197487106, 14.47355822763304125152520108341, 15.09321617520152169488056804727, 15.944480348611522676169755145119, 16.326273202276798658503608081339, 16.80124598945653692155106210441, 18.10274351781739647916198961018, 18.392702023471908635662385737434

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