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Properties

Label 1-3381-3381.818-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.508 + 0.861i$
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  + (−0.557 + 0.830i)2-s + (−0.377 − 0.925i)4-s + (−0.591 + 0.806i)5-s + (0.979 + 0.202i)8-s + (−0.339 − 0.940i)10-s + (0.301 − 0.953i)11-s + (0.992 − 0.122i)13-s + (−0.714 + 0.699i)16-s + (−0.377 + 0.925i)17-s + (0.959 + 0.281i)19-s + (0.970 + 0.242i)20-s + (0.623 + 0.781i)22-s + (−0.301 − 0.953i)25-s + (−0.452 + 0.891i)26-s + (0.377 − 0.925i)29-s + ⋯
L(s)  = 1  + (−0.557 + 0.830i)2-s + (−0.377 − 0.925i)4-s + (−0.591 + 0.806i)5-s + (0.979 + 0.202i)8-s + (−0.339 − 0.940i)10-s + (0.301 − 0.953i)11-s + (0.992 − 0.122i)13-s + (−0.714 + 0.699i)16-s + (−0.377 + 0.925i)17-s + (0.959 + 0.281i)19-s + (0.970 + 0.242i)20-s + (0.623 + 0.781i)22-s + (−0.301 − 0.953i)25-s + (−0.452 + 0.891i)26-s + (0.377 − 0.925i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.508 + 0.861i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (818, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.508 + 0.861i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4889515711 + 0.8565879324i\)
\(L(\frac12)\) \(\approx\) \(0.4889515711 + 0.8565879324i\)
\(L(1)\) \(\approx\) \(0.6599374429 + 0.3615561184i\)
\(L(1)\) \(\approx\) \(0.6599374429 + 0.3615561184i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.557 + 0.830i)T \)
5 \( 1 + (-0.591 + 0.806i)T \)
11 \( 1 + (0.301 - 0.953i)T \)
13 \( 1 + (0.992 - 0.122i)T \)
17 \( 1 + (-0.377 + 0.925i)T \)
19 \( 1 + (0.959 + 0.281i)T \)
29 \( 1 + (0.377 - 0.925i)T \)
31 \( 1 + (-0.415 + 0.909i)T \)
37 \( 1 + (0.182 + 0.983i)T \)
41 \( 1 + (-0.591 + 0.806i)T \)
43 \( 1 + (-0.979 + 0.202i)T \)
47 \( 1 + (-0.900 - 0.433i)T \)
53 \( 1 + (-0.101 - 0.994i)T \)
59 \( 1 + (0.339 + 0.940i)T \)
61 \( 1 + (0.452 + 0.891i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (0.862 - 0.505i)T \)
73 \( 1 + (-0.986 + 0.162i)T \)
79 \( 1 + (0.841 - 0.540i)T \)
83 \( 1 + (0.262 - 0.965i)T \)
89 \( 1 + (-0.768 - 0.639i)T \)
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

−18.43085828076445659253583670484, −18.15539172518047002113212313861, −17.28802589493304690690320413690, −16.60903162934820804736017446560, −15.93863887329357982264587681088, −15.43618759854996525862037162734, −14.2136549858414465725503203706, −13.53114375097433018642750877952, −12.77908147474042272221952205823, −12.26822312613979474490374288110, −11.44988802735691246519559462902, −11.1393545121764218894595223375, −10.038663646977072119386977747001, −9.31026771188214066333226177003, −8.9257671833540341140018775532, −8.055614528879283260858844494385, −7.39309674455826095814362235455, −6.688496479235764038334566721053, −5.27945326092575587306169426836, −4.69930819639512938529948679924, −3.8684944368417900117346452958, −3.26470361721634931528795641000, −2.121243808563377596193064935347, −1.3682766666287092100907351039, −0.48003359185529612233714552109, 0.874188996702209711539611259117, 1.76097432422699627999060552784, 3.13259608710726130463357072111, 3.706841481415918103465499413651, 4.66218823087122064411443351687, 5.69347119126130489207820901900, 6.33493783805408666595923884488, 6.81386502450317579308913164054, 7.8035443476757498177881230914, 8.35111906734619651655243465689, 8.851695264114521750654074677970, 10.05491164446036759138633029170, 10.39506678388817546812486404542, 11.42563268517699755870740392112, 11.62748734407042357100136725392, 13.13629695878259811768538231668, 13.62991319346322319583492862535, 14.43309632238907226391915624310, 14.97863897697845223424656669972, 15.70224983646238430958116032044, 16.23623677610202889857111382366, 16.846351352222352830127116235128, 17.874640898992554563242080338707, 18.22811381166144536971423355509, 18.94847060747488519063819136265

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