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Properties

Label 1-3089-3089.1109-r0-0-0
Degree $1$
Conductor $3089$
Sign $-0.895 - 0.444i$
Analytic cond. $14.3452$
Root an. cond. $14.3452$
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  + (−0.850 + 0.525i)2-s + (−0.253 + 0.967i)3-s + (0.447 − 0.894i)4-s + (−0.194 + 0.980i)5-s + (−0.292 − 0.956i)6-s + (−0.770 − 0.637i)7-s + (0.0894 + 0.995i)8-s + (−0.871 − 0.490i)9-s + (−0.350 − 0.936i)10-s + (−0.959 + 0.281i)11-s + (0.751 + 0.659i)12-s + (0.583 − 0.812i)13-s + (0.990 + 0.137i)14-s + (−0.899 − 0.436i)15-s + (−0.599 − 0.800i)16-s + (−0.844 + 0.535i)17-s + ⋯
L(s)  = 1  + (−0.850 + 0.525i)2-s + (−0.253 + 0.967i)3-s + (0.447 − 0.894i)4-s + (−0.194 + 0.980i)5-s + (−0.292 − 0.956i)6-s + (−0.770 − 0.637i)7-s + (0.0894 + 0.995i)8-s + (−0.871 − 0.490i)9-s + (−0.350 − 0.936i)10-s + (−0.959 + 0.281i)11-s + (0.751 + 0.659i)12-s + (0.583 − 0.812i)13-s + (0.990 + 0.137i)14-s + (−0.899 − 0.436i)15-s + (−0.599 − 0.800i)16-s + (−0.844 + 0.535i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3089\)
Sign: $-0.895 - 0.444i$
Analytic conductor: \(14.3452\)
Root analytic conductor: \(14.3452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3089} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3089,\ (0:\ ),\ -0.895 - 0.444i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08384428726 + 0.3576723428i\)
\(L(\frac12)\) \(\approx\) \(-0.08384428726 + 0.3576723428i\)
\(L(1)\) \(\approx\) \(0.4083462152 + 0.2983464301i\)
\(L(1)\) \(\approx\) \(0.4083462152 + 0.2983464301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3089 \( 1 \)
good2 \( 1 + (-0.850 + 0.525i)T \)
3 \( 1 + (-0.253 + 0.967i)T \)
5 \( 1 + (-0.194 + 0.980i)T \)
7 \( 1 + (-0.770 - 0.637i)T \)
11 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (0.583 - 0.812i)T \)
17 \( 1 + (-0.844 + 0.535i)T \)
19 \( 1 + (0.726 - 0.686i)T \)
23 \( 1 + (-0.999 + 0.0366i)T \)
29 \( 1 + (0.821 + 0.569i)T \)
31 \( 1 + (0.908 + 0.418i)T \)
37 \( 1 + (0.563 + 0.826i)T \)
41 \( 1 + (0.487 + 0.873i)T \)
43 \( 1 + (0.265 - 0.964i)T \)
47 \( 1 + (-0.358 - 0.933i)T \)
53 \( 1 + (0.921 + 0.388i)T \)
59 \( 1 + (0.137 + 0.990i)T \)
61 \( 1 + (0.504 + 0.863i)T \)
67 \( 1 + (-0.284 - 0.958i)T \)
71 \( 1 + (-0.759 - 0.650i)T \)
73 \( 1 + (0.418 + 0.908i)T \)
79 \( 1 + (0.824 + 0.566i)T \)
83 \( 1 + (0.529 - 0.848i)T \)
89 \( 1 + (-0.683 + 0.729i)T \)
97 \( 1 + (-0.997 - 0.0772i)T \)
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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

−18.5758193646642996405443806412, −18.05526736814009699809123403860, −17.4730312204951913618216381514, −16.448705736309829816076149819450, −16.06233080896839876240252988105, −15.6824963739602204693437089996, −14.049510158996938286930571416665, −13.35033903156579057295112274187, −12.8598572394266620364775496024, −12.13355925088784677963855314743, −11.68980590603122434855198322279, −11.00141531319529471220930576210, −9.89724387434529498555590570621, −9.30137434422432072214464403557, −8.48171818774427062953974112811, −8.07198438989205444717433098331, −7.28076921634773618993023650811, −6.29435846325712858869039097645, −5.81510980932279350670206210202, −4.66673097311649314203668559973, −3.66740355018031583103640577043, −2.5935605443059747313266927034, −2.10441604630590632861929297051, −1.047533219964347767592804159936, −0.22047301690118904467121628630, 0.868528144539446367870465752480, 2.50555283489028153083164222433, 3.04031228591870863800119403865, 4.01488972983786719944866716480, 4.935442504512165995583333169857, 5.86481593860959743734893782326, 6.43755362771727755811324097986, 7.15172643869865808537813687530, 8.00074708498634793623620668214, 8.66990830932712133997479904788, 9.6870158824480861770936139251, 10.30529613666941586378516495068, 10.50662464768459576536667700870, 11.23575490637449122977231919374, 12.08835363962306287009439719036, 13.43450866427213721096044438968, 13.882530413176196521406753550158, 15.01399589088288971255147462131, 15.35633121455573071803107438327, 15.94482905950960663878941708873, 16.41622367456515240610444105524, 17.457347466836025593645848836595, 17.9398570233860838586582826672, 18.38702996797266437487093911491, 19.62311323377036396106403901507

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