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Properties

Label 1-3381-3381.431-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.302 - 0.953i$
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  + (0.403 + 0.915i)2-s + (−0.675 + 0.737i)4-s + (−0.275 + 0.961i)5-s + (−0.947 − 0.320i)8-s + (−0.990 + 0.135i)10-s + (−0.0339 − 0.999i)11-s + (0.488 + 0.872i)13-s + (−0.0882 − 0.996i)16-s + (0.976 − 0.215i)17-s + (0.786 + 0.618i)19-s + (−0.523 − 0.852i)20-s + (0.900 − 0.433i)22-s + (−0.848 − 0.529i)25-s + (−0.601 + 0.798i)26-s + (0.301 − 0.953i)29-s + ⋯
L(s)  = 1  + (0.403 + 0.915i)2-s + (−0.675 + 0.737i)4-s + (−0.275 + 0.961i)5-s + (−0.947 − 0.320i)8-s + (−0.990 + 0.135i)10-s + (−0.0339 − 0.999i)11-s + (0.488 + 0.872i)13-s + (−0.0882 − 0.996i)16-s + (0.976 − 0.215i)17-s + (0.786 + 0.618i)19-s + (−0.523 − 0.852i)20-s + (0.900 − 0.433i)22-s + (−0.848 − 0.529i)25-s + (−0.601 + 0.798i)26-s + (0.301 − 0.953i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.302 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.302 - 0.953i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.302 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.002021425803 + 0.001479289643i\)
\(L(\frac12)\) \(\approx\) \(0.002021425803 + 0.001479289643i\)
\(L(1)\) \(\approx\) \(0.7488518875 + 0.5709282866i\)
\(L(1)\) \(\approx\) \(0.7488518875 + 0.5709282866i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.403 + 0.915i)T \)
5 \( 1 + (-0.275 + 0.961i)T \)
11 \( 1 + (-0.0339 - 0.999i)T \)
13 \( 1 + (0.488 + 0.872i)T \)
17 \( 1 + (0.976 - 0.215i)T \)
19 \( 1 + (0.786 + 0.618i)T \)
29 \( 1 + (0.301 - 0.953i)T \)
31 \( 1 + (-0.888 - 0.458i)T \)
37 \( 1 + (-0.855 - 0.517i)T \)
41 \( 1 + (-0.970 + 0.242i)T \)
43 \( 1 + (-0.947 + 0.320i)T \)
47 \( 1 + (-0.955 - 0.294i)T \)
53 \( 1 + (-0.634 - 0.773i)T \)
59 \( 1 + (-0.990 + 0.135i)T \)
61 \( 1 + (-0.390 + 0.920i)T \)
67 \( 1 + (0.327 - 0.945i)T \)
71 \( 1 + (-0.917 - 0.396i)T \)
73 \( 1 + (0.155 + 0.987i)T \)
79 \( 1 + (-0.235 + 0.971i)T \)
83 \( 1 + (-0.979 + 0.202i)T \)
89 \( 1 + (-0.248 - 0.968i)T \)
97 \( 1 + (-0.415 + 0.909i)T \)
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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

−19.11849070843136491663894669803, −18.317084469387284020032255662285, −17.72959785396652973144485246809, −17.02909835324007263041590811346, −16.069891749037841702831201217625, −15.41829253801976357994996883860, −14.77408926269221196597041838462, −13.89032762735654885690111523733, −13.22685641101576777626049349099, −12.51284186067241256759426991047, −12.20975341074982629456333098783, −11.39664160482473344864631301459, −10.53176540306237595566242599683, −9.92121025746184223949499205601, −9.20361490872175825556413159505, −8.501484196629580120595421367654, −7.74088748040192335837846397912, −6.76661254778210074570579433149, −5.58178804786538978456826427128, −5.137847423187486066010617118782, −4.5142305458046006189604593264, −3.47607052177174322416332863101, −3.04815770036641715712282010506, −1.62740339125276917186234404191, −1.31105272657484278691199846026, 0.00064240285949231585541873128, 1.517192082695195620395231874083, 2.86381907349578191641977110187, 3.48047455615291116265492819493, 4.01006428139559106044457634847, 5.1520599904948840683868200612, 5.86176642045701995780304059908, 6.47824069804203879999060503815, 7.182480074676323917384453130104, 7.929785398510424853427879826210, 8.46488299178244802775313628377, 9.47045102059209015288153186946, 10.10233283039331430033354098260, 11.21076872846512410538229886988, 11.68461045878604504080175314891, 12.41669176000880034706613995767, 13.62228953431117373999324604155, 13.80259895769778741837760559131, 14.56556963705923148226746208864, 15.119843508842559723241376781035, 16.08592037802987930066741859964, 16.33715926737198096039667036238, 17.12636022833415822012083560869, 18.067330686337557653955529407532, 18.69987458146244030558426047522

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