Properties

Label 1-3381-3381.353-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.212 + 0.977i$
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  + (0.833 + 0.552i)2-s + (0.390 + 0.920i)4-s + (−0.00679 − 0.999i)5-s + (−0.182 + 0.983i)8-s + (0.546 − 0.837i)10-s + (−0.511 + 0.859i)11-s + (0.979 + 0.202i)13-s + (−0.694 + 0.719i)16-s + (0.601 + 0.798i)17-s + (0.888 − 0.458i)19-s + (0.917 − 0.396i)20-s + (−0.900 + 0.433i)22-s + (−0.999 + 0.0135i)25-s + (0.704 + 0.709i)26-s + (0.992 − 0.122i)29-s + ⋯
L(s)  = 1  + (0.833 + 0.552i)2-s + (0.390 + 0.920i)4-s + (−0.00679 − 0.999i)5-s + (−0.182 + 0.983i)8-s + (0.546 − 0.837i)10-s + (−0.511 + 0.859i)11-s + (0.979 + 0.202i)13-s + (−0.694 + 0.719i)16-s + (0.601 + 0.798i)17-s + (0.888 − 0.458i)19-s + (0.917 − 0.396i)20-s + (−0.900 + 0.433i)22-s + (−0.999 + 0.0135i)25-s + (0.704 + 0.709i)26-s + (0.992 − 0.122i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.212 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.212 + 0.977i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.212 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.339012294 + 1.885566386i\)
\(L(\frac12)\) \(\approx\) \(2.339012294 + 1.885566386i\)
\(L(1)\) \(\approx\) \(1.655841240 + 0.6309526347i\)
\(L(1)\) \(\approx\) \(1.655841240 + 0.6309526347i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.833 + 0.552i)T \)
5 \( 1 + (-0.00679 - 0.999i)T \)
11 \( 1 + (-0.511 + 0.859i)T \)
13 \( 1 + (0.979 + 0.202i)T \)
17 \( 1 + (0.601 + 0.798i)T \)
19 \( 1 + (0.888 - 0.458i)T \)
29 \( 1 + (0.992 - 0.122i)T \)
31 \( 1 + (0.327 - 0.945i)T \)
37 \( 1 + (-0.675 - 0.737i)T \)
41 \( 1 + (-0.862 + 0.505i)T \)
43 \( 1 + (0.182 + 0.983i)T \)
47 \( 1 + (0.955 + 0.294i)T \)
53 \( 1 + (0.169 - 0.985i)T \)
59 \( 1 + (-0.546 + 0.837i)T \)
61 \( 1 + (0.966 + 0.255i)T \)
67 \( 1 + (0.235 + 0.971i)T \)
71 \( 1 + (-0.986 + 0.162i)T \)
73 \( 1 + (-0.963 - 0.268i)T \)
79 \( 1 + (0.580 + 0.814i)T \)
83 \( 1 + (0.996 + 0.0815i)T \)
89 \( 1 + (0.994 - 0.108i)T \)
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

−18.74726995867354592344367609444, −18.34963264751907866122034199738, −17.46258353998813726325853600134, −16.14017891969566407393800263942, −15.88677495654109234473176423969, −15.145275214998391434705195659911, −14.22351597570098222546590272363, −13.76191954117279848813154871119, −13.44623613744475636085020771359, −12.03077465414224460750520608751, −11.96554626353488167306193657095, −10.80236024496982562094530125146, −10.59225162865579277814184014178, −9.83889848114497159051248855925, −8.83935331505077717056337681660, −7.90074086845568428287467492175, −7.03363028810602236467429600004, −6.35979495966184325315821452103, −5.601172788292076412405796135713, −5.04488225189697230168920476603, −3.82460746753882196771469558451, −3.21489669115013057761760573816, −2.81981132943643449951787806366, −1.68085418059604513877588355975, −0.727968275787152330927734630178, 1.07762170900489842814297619172, 2.01331585530306694798550143765, 3.00202801082167409966711369139, 3.95275147574972122317578889033, 4.49939968463531567210402616931, 5.31200437797386399687674416622, 5.85497422203314551382451827809, 6.72646441861863662023024298953, 7.60687360626016474454016642989, 8.18639326523291267907811485268, 8.85813026063771187278232041084, 9.740322996773805096410187285640, 10.624842284612750372571953023409, 11.6630801600149890948723746183, 12.083026466447981004334030955771, 12.94042997602994352102241442083, 13.29990861104608242324722613899, 14.03635160314524647522389558873, 14.891334146402975515332305067340, 15.59390634395415197006439608686, 16.09301901169266035886261910748, 16.67681652449374797684156258281, 17.57391747515635928511788885171, 17.88435595212258028913807095637, 19.03449437652481029691076216395

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