Properties

Label 1-3381-3381.296-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.173 - 0.984i$
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  + (0.951 + 0.307i)2-s + (0.810 + 0.585i)4-s + (−0.427 − 0.903i)5-s + (0.591 + 0.806i)8-s + (−0.128 − 0.991i)10-s + (−0.352 + 0.935i)11-s + (−0.768 − 0.639i)13-s + (0.314 + 0.949i)16-s + (−0.912 − 0.409i)17-s + (0.888 + 0.458i)19-s + (0.182 − 0.983i)20-s + (−0.623 + 0.781i)22-s + (−0.634 + 0.773i)25-s + (−0.534 − 0.844i)26-s + (−0.101 − 0.994i)29-s + ⋯
L(s)  = 1  + (0.951 + 0.307i)2-s + (0.810 + 0.585i)4-s + (−0.427 − 0.903i)5-s + (0.591 + 0.806i)8-s + (−0.128 − 0.991i)10-s + (−0.352 + 0.935i)11-s + (−0.768 − 0.639i)13-s + (0.314 + 0.949i)16-s + (−0.912 − 0.409i)17-s + (0.888 + 0.458i)19-s + (0.182 − 0.983i)20-s + (−0.623 + 0.781i)22-s + (−0.634 + 0.773i)25-s + (−0.534 − 0.844i)26-s + (−0.101 − 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.173 - 0.984i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.173 - 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9467579240 - 1.128552143i\)
\(L(\frac12)\) \(\approx\) \(0.9467579240 - 1.128552143i\)
\(L(1)\) \(\approx\) \(1.420187251 + 0.02407765900i\)
\(L(1)\) \(\approx\) \(1.420187251 + 0.02407765900i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.951 + 0.307i)T \)
5 \( 1 + (-0.427 - 0.903i)T \)
11 \( 1 + (-0.352 + 0.935i)T \)
13 \( 1 + (-0.768 - 0.639i)T \)
17 \( 1 + (-0.912 - 0.409i)T \)
19 \( 1 + (0.888 + 0.458i)T \)
29 \( 1 + (-0.101 - 0.994i)T \)
31 \( 1 + (-0.327 - 0.945i)T \)
37 \( 1 + (-0.869 - 0.494i)T \)
41 \( 1 + (-0.996 - 0.0815i)T \)
43 \( 1 + (0.591 - 0.806i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (0.998 + 0.0543i)T \)
59 \( 1 + (-0.128 - 0.991i)T \)
61 \( 1 + (-0.464 - 0.885i)T \)
67 \( 1 + (-0.235 + 0.971i)T \)
71 \( 1 + (0.377 - 0.925i)T \)
73 \( 1 + (0.390 - 0.920i)T \)
79 \( 1 + (-0.580 + 0.814i)T \)
83 \( 1 + (0.557 - 0.830i)T \)
89 \( 1 + (0.704 - 0.709i)T \)
97 \( 1 + (0.142 + 0.989i)T \)
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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

−19.287543104068876855510332023919, −18.446050071545260165446784944426, −17.82561790574108140264439796593, −16.658525548510013428706125812766, −16.071384909045544328115013000312, −15.40021188566182174759930869072, −14.78345300658497593324508470506, −14.062742050956843209576114130375, −13.63527191814789681602499420995, −12.76154445280894952679428594064, −11.94179184768793340889688918351, −11.400708025805613440010630795091, −10.77873694676106151914430818278, −10.2159779780829957520865035965, −9.23414803301098568829074742878, −8.295127629276829291984129162026, −7.23008040345266905340760726352, −6.88844527142450630370863079801, −6.074395439499145574100990850278, −5.19603534503500309564071729429, −4.511323968712665486428975131796, −3.55943488072686412014938742434, −3.024809879109648654676668757067, −2.29976659166256685045179925277, −1.269741263406571523587182395908, 0.27962724606598216819370932276, 1.77988823731242927077337803386, 2.41458265448700623745634722777, 3.50496019495895161566241959862, 4.1862372087701782842924178738, 5.06714383682481440840347455955, 5.27099909906246791024331817912, 6.357899777557503961309829376296, 7.38138697565678161246127628044, 7.643308251647937286021419937565, 8.520121529046873437875248771584, 9.4578793730692707390761057196, 10.21301134883788829328497333744, 11.19427315427991971692205200157, 12.017919210675458948106791158574, 12.28606526355762441710173613526, 13.18618814931504269060982719754, 13.543030502261066804700618746405, 14.57037294417946592388533169566, 15.30289612455195981148299255366, 15.65137106070001865013568137878, 16.38864603987041827486073854756, 17.2072456903306558508029851748, 17.59081175435874015525776400962, 18.63480409989863672225967352977

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