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Properties

Label 1-3381-3381.1850-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: $\chi_{3381} (1850, \cdot )$
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

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

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