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Properties

Label 1-3381-3381.2573-r0-0-0
Degree $1$
Conductor $3381$
Sign $-0.468 + 0.883i$
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  + (0.634 − 0.773i)2-s + (−0.195 − 0.980i)4-s + (0.155 − 0.987i)5-s + (−0.882 − 0.470i)8-s + (−0.665 − 0.746i)10-s + (0.209 − 0.977i)11-s + (0.0203 − 0.999i)13-s + (−0.923 + 0.384i)16-s + (−0.751 − 0.659i)17-s + (−0.0475 − 0.998i)19-s + (−0.999 + 0.0407i)20-s + (−0.623 − 0.781i)22-s + (−0.951 − 0.307i)25-s + (−0.760 − 0.649i)26-s + (−0.947 + 0.320i)29-s + ⋯
L(s)  = 1  + (0.634 − 0.773i)2-s + (−0.195 − 0.980i)4-s + (0.155 − 0.987i)5-s + (−0.882 − 0.470i)8-s + (−0.665 − 0.746i)10-s + (0.209 − 0.977i)11-s + (0.0203 − 0.999i)13-s + (−0.923 + 0.384i)16-s + (−0.751 − 0.659i)17-s + (−0.0475 − 0.998i)19-s + (−0.999 + 0.0407i)20-s + (−0.623 − 0.781i)22-s + (−0.951 − 0.307i)25-s + (−0.760 − 0.649i)26-s + (−0.947 + 0.320i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $-0.468 + 0.883i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (2573, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ -0.468 + 0.883i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.8961396749 - 1.489856265i\)
\(L(\frac12)\) \(\approx\) \(-0.8961396749 - 1.489856265i\)
\(L(1)\) \(\approx\) \(0.7366281675 - 1.077545446i\)
\(L(1)\) \(\approx\) \(0.7366281675 - 1.077545446i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.634 - 0.773i)T \)
5 \( 1 + (0.155 - 0.987i)T \)
11 \( 1 + (0.209 - 0.977i)T \)
13 \( 1 + (0.0203 - 0.999i)T \)
17 \( 1 + (-0.751 - 0.659i)T \)
19 \( 1 + (-0.0475 - 0.998i)T \)
29 \( 1 + (-0.947 + 0.320i)T \)
31 \( 1 + (0.981 + 0.189i)T \)
37 \( 1 + (0.973 - 0.229i)T \)
41 \( 1 + (0.933 - 0.359i)T \)
43 \( 1 + (-0.882 + 0.470i)T \)
47 \( 1 + (-0.826 + 0.563i)T \)
53 \( 1 + (-0.694 - 0.719i)T \)
59 \( 1 + (-0.665 - 0.746i)T \)
61 \( 1 + (0.942 + 0.333i)T \)
67 \( 1 + (-0.723 - 0.690i)T \)
71 \( 1 + (0.818 + 0.574i)T \)
73 \( 1 + (0.999 + 0.0271i)T \)
79 \( 1 + (0.995 + 0.0950i)T \)
83 \( 1 + (-0.301 - 0.953i)T \)
89 \( 1 + (-0.802 + 0.596i)T \)
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

−19.179828801569220427504167921371, −18.33681492507561626686157769495, −17.9204727572770145884617425352, −16.98439748131881952334651512031, −16.66629809666246603016912829138, −15.503947359566391461925139902609, −15.14521197505968852339644970884, −14.48171503824631403073063438228, −13.935766149384402871725868275302, −13.175704012298669474451812123123, −12.451201806363875712010447133712, −11.6499649446940745335061764983, −11.10607908655862624884373965751, −10.00176569485963116915753070937, −9.458878184280675107215286695797, −8.466731027618693092750282570185, −7.68556567476228467605634900162, −7.040867273369970316704835427975, −6.35054321423145989368296450114, −5.93356150908721657326610182781, −4.72972073363293127404158937182, −4.14092192521862688207228197744, −3.44912456565183828029289116871, −2.38992842780481415757928870118, −1.78614339848561652607751979120, 0.426614636668664739828234340844, 1.04614656068401626503580944127, 2.137223449372517965045924758785, 2.940254301093693667713916222447, 3.70362992273734736399012256221, 4.68852839183250211097196021212, 5.101800083951977795806741707891, 5.9432367536450912992794446773, 6.58938915554512460982412637530, 7.83587644733115682262930804395, 8.627432787768431990773347977321, 9.30654152588034268138605087141, 9.85009542260485723687901412150, 10.99834522946334837071503863703, 11.25407309022759495683300857106, 12.135387183841373532276522291149, 12.98523096290549845981157832517, 13.23697755851885731295468929195, 13.95247977453392526283689603390, 14.76862317224681718227345452071, 15.65417519001338524318763563397, 16.05604682307610264676350861984, 17.03076695341899488085594460635, 17.80096172717798504981817560986, 18.368243424258570297720219075578

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