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Properties

Label 1-1027-1027.710-r0-0-0
Degree $1$
Conductor $1027$
Sign $-0.289 - 0.957i$
Analytic cond. $4.76936$
Root an. cond. $4.76936$
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  + i·2-s − 3-s − 4-s + i·5-s − i·6-s + i·7-s − i·8-s + 9-s − 10-s − i·11-s + 12-s − 14-s − i·15-s + 16-s + 17-s + i·18-s + ⋯
L(s)  = 1  + i·2-s − 3-s − 4-s + i·5-s − i·6-s + i·7-s − i·8-s + 9-s − 10-s − i·11-s + 12-s − 14-s − i·15-s + 16-s + 17-s + i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $-0.289 - 0.957i$
Analytic conductor: \(4.76936\)
Root analytic conductor: \(4.76936\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1027} (710, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1027,\ (0:\ ),\ -0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2422224665 + 0.3264207507i\)
\(L(\frac12)\) \(\approx\) \(-0.2422224665 + 0.3264207507i\)
\(L(1)\) \(\approx\) \(0.4095963894 + 0.4599700051i\)
\(L(1)\) \(\approx\) \(0.4095963894 + 0.4599700051i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
79 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + iT \)
7 \( 1 - T \)
11 \( 1 - T \)
17 \( 1 - iT \)
19 \( 1 + iT \)
23 \( 1 - iT \)
29 \( 1 + T \)
31 \( 1 - T \)
37 \( 1 - iT \)
41 \( 1 + T \)
43 \( 1 \)
47 \( 1 - T \)
53 \( 1 - iT \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + iT \)
71 \( 1 + iT \)
73 \( 1 - iT \)
83 \( 1 + T \)
89 \( 1 - T \)
97 \( 1 + iT \)
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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

−20.96123186256391004951481435137, −20.2970037199496067771388737717, −19.74145735525456860096289387875, −18.6924853392285120047315428001, −17.80904268793153432281741597020, −17.22221047624480028674222062223, −16.69289959766587258415719676778, −15.746686969430792939681525957622, −14.55047287243532380558671624958, −13.497347108459321041532302246808, −12.867632931996497860614484831895, −12.27967395067262494998635610799, −11.48012812844343068051012315751, −10.73855589518794289382755444437, −9.752560009606910514569392564411, −9.483482098921627830558693682277, −7.9979830105792325763033830774, −7.32684410452773365602749758327, −5.95348434963936241326686578599, −5.06046603313242314686161582845, −4.38506206858392515684900073822, −3.74675997964592752789786381101, −2.05848864322952149153331111713, −1.19480497410937830648454443465, −0.22699630115796122879837937187, 1.4941976717759300366027283005, 3.10703094251998038267975128830, 3.98534330776446608430545455285, 5.288691630900300330251739716561, 5.92929538928359680995942772114, 6.29284264324948038380197095171, 7.440551394663105625577683191011, 8.09892938378656262627132563586, 9.2618235922152787573784247611, 10.11718168389799132909443677441, 10.87480191272503441608571075653, 11.9458936824497306046919853407, 12.47826336942322086166432704128, 13.69590945988982764689036309585, 14.396199295243716181806161912574, 15.18568683349599901122162387451, 16.000670847587882563578147665301, 16.48240188672338096726988870229, 17.46827883091420878702128117420, 18.1721671362129215109011213513, 18.844274399212069514033400973693, 19.0848033287710941968775195331, 21.06064457271699352962418753314, 21.683934653870832963017179095483, 22.426350932102894939278033289447

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