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Properties

Label 1-4013-4013.530-r0-0-0
Degree $1$
Conductor $4013$
Sign $0.982 - 0.185i$
Analytic cond. $18.6362$
Root an. cond. $18.6362$
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  + (0.982 + 0.183i)2-s + (−0.445 + 0.895i)3-s + (0.932 + 0.361i)4-s + (0.850 − 0.526i)5-s + (−0.602 + 0.798i)6-s + (0.445 − 0.895i)7-s + (0.850 + 0.526i)8-s + (−0.602 − 0.798i)9-s + (0.932 − 0.361i)10-s + (0.739 − 0.673i)11-s + (−0.739 + 0.673i)12-s + (0.739 − 0.673i)13-s + (0.602 − 0.798i)14-s + (0.0922 + 0.995i)15-s + (0.739 + 0.673i)16-s + (0.445 + 0.895i)17-s + ⋯
L(s)  = 1  + (0.982 + 0.183i)2-s + (−0.445 + 0.895i)3-s + (0.932 + 0.361i)4-s + (0.850 − 0.526i)5-s + (−0.602 + 0.798i)6-s + (0.445 − 0.895i)7-s + (0.850 + 0.526i)8-s + (−0.602 − 0.798i)9-s + (0.932 − 0.361i)10-s + (0.739 − 0.673i)11-s + (−0.739 + 0.673i)12-s + (0.739 − 0.673i)13-s + (0.602 − 0.798i)14-s + (0.0922 + 0.995i)15-s + (0.739 + 0.673i)16-s + (0.445 + 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4013\)
Sign: $0.982 - 0.185i$
Analytic conductor: \(18.6362\)
Root analytic conductor: \(18.6362\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4013} (530, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4013,\ (0:\ ),\ 0.982 - 0.185i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.212628761 - 0.3931514304i\)
\(L(\frac12)\) \(\approx\) \(4.212628761 - 0.3931514304i\)
\(L(1)\) \(\approx\) \(2.229582756 + 0.1750819470i\)
\(L(1)\) \(\approx\) \(2.229582756 + 0.1750819470i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4013 \( 1 \)
good2 \( 1 + (0.982 + 0.183i)T \)
3 \( 1 + (-0.445 + 0.895i)T \)
5 \( 1 + (0.850 - 0.526i)T \)
7 \( 1 + (0.445 - 0.895i)T \)
11 \( 1 + (0.739 - 0.673i)T \)
13 \( 1 + (0.739 - 0.673i)T \)
17 \( 1 + (0.445 + 0.895i)T \)
19 \( 1 + (-0.602 - 0.798i)T \)
23 \( 1 + (-0.0922 - 0.995i)T \)
29 \( 1 + (0.273 - 0.961i)T \)
31 \( 1 + (-0.602 + 0.798i)T \)
37 \( 1 + (0.602 + 0.798i)T \)
41 \( 1 + (0.0922 + 0.995i)T \)
43 \( 1 + (0.445 + 0.895i)T \)
47 \( 1 + (-0.850 - 0.526i)T \)
53 \( 1 + (-0.982 - 0.183i)T \)
59 \( 1 + (0.0922 - 0.995i)T \)
61 \( 1 + (-0.602 - 0.798i)T \)
67 \( 1 + (-0.982 - 0.183i)T \)
71 \( 1 + (0.0922 + 0.995i)T \)
73 \( 1 + (0.0922 + 0.995i)T \)
79 \( 1 + (-0.739 - 0.673i)T \)
83 \( 1 + (-0.602 + 0.798i)T \)
89 \( 1 + (-0.982 - 0.183i)T \)
97 \( 1 + (-0.982 + 0.183i)T \)
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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

−18.4229318243607807244890112531, −18.030560050111972505131777474645, −17.13120226535264302738449962337, −16.50560864786973787461769090158, −15.67356281147438907980479682119, −14.69308631237857296294828973371, −14.342659949230512042508803924817, −13.75987703488511960936994899089, −13.00938544899122173970485480970, −12.31179547923294693958454204518, −11.80203156662589220662903935088, −11.17979631307308006937117122295, −10.54921885112890345721281003579, −9.51770964735652263655919113774, −8.864588364594769201947060308728, −7.57872801066533338008128747861, −7.15227669178130368793423819203, −6.229890377252579818825303071686, −5.88769017119359344201046984661, −5.25217577279791995412622027074, −4.34413206609038246197626805132, −3.309710902246804010826177660213, −2.4375473952908311845300080178, −1.73323072798784791831763191178, −1.41415060739207653202315514833, 0.8668090079138697642249878832, 1.62395085903189373317307924249, 2.9061638803158712376133299763, 3.57653021159780942841211682083, 4.49072637783881430872420614644, 4.70767308680553578540854036744, 5.80737900954373845157773024193, 6.170765855337885349956546540751, 6.80546087357845049953269801500, 8.28521612237385945367314976608, 8.42932801853479865484569518225, 9.70645435246493457934385004910, 10.30833470996670092197303367026, 11.09113048330230787023982119097, 11.33781935638758043621181750868, 12.53197914288361748830908332479, 12.982665667576219664110394742831, 13.79735556933986257704564166015, 14.3884170173481001898978448683, 14.895742286970942081109605031506, 15.78954427073404746045383488911, 16.47709215648704671406593188549, 16.93424893575126631302228129910, 17.37815139741858880610590407073, 18.125782293009532016400798304633

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