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Properties

Label 1-3381-3381.53-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.487 - 0.873i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (−0.869 + 0.494i)2-s + (0.511 − 0.859i)4-s + (−0.115 + 0.993i)5-s + (−0.0203 + 0.999i)8-s + (−0.390 − 0.920i)10-s + (0.288 − 0.957i)11-s + (0.947 − 0.320i)13-s + (−0.476 − 0.879i)16-s + (−0.999 + 0.0135i)17-s + (−0.235 + 0.971i)19-s + (0.794 + 0.607i)20-s + (0.222 + 0.974i)22-s + (−0.973 − 0.229i)25-s + (−0.665 + 0.746i)26-s + (−0.488 + 0.872i)29-s + ⋯
L(s)  = 1  + (−0.869 + 0.494i)2-s + (0.511 − 0.859i)4-s + (−0.115 + 0.993i)5-s + (−0.0203 + 0.999i)8-s + (−0.390 − 0.920i)10-s + (0.288 − 0.957i)11-s + (0.947 − 0.320i)13-s + (−0.476 − 0.879i)16-s + (−0.999 + 0.0135i)17-s + (−0.235 + 0.971i)19-s + (0.794 + 0.607i)20-s + (0.222 + 0.974i)22-s + (−0.973 − 0.229i)25-s + (−0.665 + 0.746i)26-s + (−0.488 + 0.872i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.487 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.487 - 0.873i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.487 - 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5089709491 - 0.2987317741i\)
\(L(\frac12)\) \(\approx\) \(0.5089709491 - 0.2987317741i\)
\(L(1)\) \(\approx\) \(0.6378875727 + 0.1295184875i\)
\(L(1)\) \(\approx\) \(0.6378875727 + 0.1295184875i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.869 + 0.494i)T \)
5 \( 1 + (-0.115 + 0.993i)T \)
11 \( 1 + (0.288 - 0.957i)T \)
13 \( 1 + (0.947 - 0.320i)T \)
17 \( 1 + (-0.999 + 0.0135i)T \)
19 \( 1 + (-0.235 + 0.971i)T \)
29 \( 1 + (-0.488 + 0.872i)T \)
31 \( 1 + (0.580 - 0.814i)T \)
37 \( 1 + (0.0339 - 0.999i)T \)
41 \( 1 + (-0.917 + 0.396i)T \)
43 \( 1 + (-0.0203 - 0.999i)T \)
47 \( 1 + (-0.365 - 0.930i)T \)
53 \( 1 + (-0.248 + 0.968i)T \)
59 \( 1 + (-0.390 - 0.920i)T \)
61 \( 1 + (-0.314 + 0.949i)T \)
67 \( 1 + (0.786 - 0.618i)T \)
71 \( 1 + (0.933 - 0.359i)T \)
73 \( 1 + (-0.0882 + 0.996i)T \)
79 \( 1 + (0.888 - 0.458i)T \)
83 \( 1 + (0.182 - 0.983i)T \)
89 \( 1 + (-0.275 + 0.961i)T \)
97 \( 1 + (0.654 - 0.755i)T \)
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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

−19.01988699587746552083124724174, −18.16202092733595048503418848220, −17.507508749435364244511158429319, −17.12077363136017569788185377887, −16.26993677617501412857194118591, −15.628803751715164284150392718002, −15.173360717062042733438851786956, −13.79120259725303999689198136905, −13.15859811058055359351334414425, −12.62810973147667819482061059965, −11.75104544459535228145035445855, −11.3408127543202067623914778282, −10.47626734235000766018774751071, −9.54967796285738130190186178460, −9.166056827622866759851795388595, −8.41467732919611569095634678255, −7.87193197996015315443703318742, −6.7825894979015861408818632808, −6.38009755553340988070070950064, −4.97006119278380659973352747436, −4.37451727790034053529286954738, −3.61506475659205312597891939723, −2.49045001925075660894555869247, −1.709904487777600638018295409729, −0.99614878110161214069022232856, 0.2615496114464550901281228350, 1.48051740166081961811922489626, 2.30481739028901023223148119012, 3.30176095291009380921410399397, 3.9936284850580033748574316786, 5.33337653972508002423150431573, 6.08611760770382340229938119233, 6.518148064160128958251285778484, 7.30969920401058796596119386985, 8.15442724662386115008720719813, 8.639788063384256239884854433639, 9.47371277202757408344569733096, 10.334702903482223080688770179669, 10.948509847018220026782049133643, 11.27659155505315743278609921525, 12.23311617117807882844151148913, 13.49443133741994806191306596978, 13.92144706645345868379357335320, 14.777513120377540532067930873284, 15.31067507949225282113127727101, 15.98572469965931406772156819332, 16.64877438779285189133267413595, 17.36592552718516756437509672951, 18.17679653755588048888889556664, 18.60851354329048170201446001225

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