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Properties

Label 1-3381-3381.1040-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.556 + 0.830i$
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  + (−0.601 + 0.798i)2-s + (−0.275 − 0.961i)4-s + (0.511 + 0.859i)5-s + (0.933 + 0.359i)8-s + (−0.994 − 0.108i)10-s + (0.999 + 0.0271i)11-s + (0.917 + 0.396i)13-s + (−0.848 + 0.529i)16-s + (−0.694 − 0.719i)17-s + (0.995 + 0.0950i)19-s + (0.685 − 0.728i)20-s + (−0.623 + 0.781i)22-s + (−0.476 + 0.879i)25-s + (−0.869 + 0.494i)26-s + (−0.970 + 0.242i)29-s + ⋯
L(s)  = 1  + (−0.601 + 0.798i)2-s + (−0.275 − 0.961i)4-s + (0.511 + 0.859i)5-s + (0.933 + 0.359i)8-s + (−0.994 − 0.108i)10-s + (0.999 + 0.0271i)11-s + (0.917 + 0.396i)13-s + (−0.848 + 0.529i)16-s + (−0.694 − 0.719i)17-s + (0.995 + 0.0950i)19-s + (0.685 − 0.728i)20-s + (−0.623 + 0.781i)22-s + (−0.476 + 0.879i)25-s + (−0.869 + 0.494i)26-s + (−0.970 + 0.242i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.556 + 0.830i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1040, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.556 + 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6742887074 + 1.263794112i\)
\(L(\frac12)\) \(\approx\) \(0.6742887074 + 1.263794112i\)
\(L(1)\) \(\approx\) \(0.8066931626 + 0.4958729322i\)
\(L(1)\) \(\approx\) \(0.8066931626 + 0.4958729322i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.601 + 0.798i)T \)
5 \( 1 + (0.511 + 0.859i)T \)
11 \( 1 + (0.999 + 0.0271i)T \)
13 \( 1 + (0.917 + 0.396i)T \)
17 \( 1 + (-0.694 - 0.719i)T \)
19 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (-0.970 + 0.242i)T \)
31 \( 1 + (0.928 - 0.371i)T \)
37 \( 1 + (-0.906 + 0.421i)T \)
41 \( 1 + (-0.488 + 0.872i)T \)
43 \( 1 + (0.933 - 0.359i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (0.760 - 0.649i)T \)
59 \( 1 + (-0.994 - 0.108i)T \)
61 \( 1 + (0.00679 + 0.999i)T \)
67 \( 1 + (-0.0475 + 0.998i)T \)
71 \( 1 + (-0.947 + 0.320i)T \)
73 \( 1 + (-0.876 + 0.482i)T \)
79 \( 1 + (-0.981 + 0.189i)T \)
83 \( 1 + (0.986 + 0.162i)T \)
89 \( 1 + (-0.675 - 0.737i)T \)
97 \( 1 + (0.959 + 0.281i)T \)
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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

−18.58542712924929774671446183822, −17.81484321068192044878603615835, −17.3422792388258720522679917540, −16.76854377452913911398080133678, −16.006253373477444460374632619154, −15.34172987180786405206738693550, −14.04965235548794338296528863167, −13.594280056811808732271127606363, −12.92687698832351727269539273857, −12.18361767170442908506792850014, −11.6478300665654941623840792755, −10.77278526089125009375603435140, −10.1734526234406442284653567211, −9.20915822381492360209272960721, −8.945465930495329148004663660, −8.23215222794643334844869839226, −7.35572612746543492199786164386, −6.38792140799320669665007890075, −5.6043655349574666719815706696, −4.61311944605958838788797632268, −3.88825495692774253103465133539, −3.18849914269975682592244372388, −1.97895818108159707908324912949, −1.461534474002365092222470291618, −0.5889479147222904500058595949, 1.085585252810170618961811926344, 1.77907875782322920641658469895, 2.85232841150347665408457766075, 3.84597159154956194298363419922, 4.72395140749618228777802039595, 5.75933492238547006147188678853, 6.22979552496916231136941128178, 7.00260126344179781023482800437, 7.422209194905505608407708671817, 8.55929272001640631766335672741, 9.14558222917035454922108006461, 9.75450339317441358086682832637, 10.459868873141173824743912274373, 11.35851782816070379398592354792, 11.67884792973652318027158287021, 13.200407332729744182762020613380, 13.74189419155525311027887164198, 14.28364868168276349717649858103, 14.91963124292370554854712097681, 15.725843095399977347453261788387, 16.238693279517772198896120243148, 17.155553422596400550204891894073, 17.60882957363477716090954723593, 18.37329614751134231614380401110, 18.763111294904066505291123597838

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