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Properties

Label 1-1011-1011.485-r1-0-0
Degree $1$
Conductor $1011$
Sign $-0.504 - 0.863i$
Analytic cond. $108.647$
Root an. cond. $108.647$
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  − 2-s + 4-s − i·5-s − 7-s − 8-s + i·10-s + i·11-s + 13-s + 14-s + 16-s − i·17-s + i·19-s − i·20-s − i·22-s − i·23-s + ⋯
L(s)  = 1  − 2-s + 4-s − i·5-s − 7-s − 8-s + i·10-s + i·11-s + 13-s + 14-s + 16-s − i·17-s + i·19-s − i·20-s − i·22-s − i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.504 - 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1011\)    =    \(3 \cdot 337\)
Sign: $-0.504 - 0.863i$
Analytic conductor: \(108.647\)
Root analytic conductor: \(108.647\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1011} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1011,\ (1:\ ),\ -0.504 - 0.863i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3644556604 - 0.6353609560i\)
\(L(\frac12)\) \(\approx\) \(0.3644556604 - 0.6353609560i\)
\(L(1)\) \(\approx\) \(0.6114925626 - 0.1287075950i\)
\(L(1)\) \(\approx\) \(0.6114925626 - 0.1287075950i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
337 \( 1 \)
good2 \( 1 \)
5 \( 1 - T \)
7 \( 1 \)
11 \( 1 + T \)
13 \( 1 - iT \)
17 \( 1 \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 \)
31 \( 1 + iT \)
37 \( 1 + iT \)
41 \( 1 \)
43 \( 1 + T \)
47 \( 1 + T \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 \)
71 \( 1 + iT \)
73 \( 1 - iT \)
79 \( 1 \)
83 \( 1 - iT \)
89 \( 1 - iT \)
97 \( 1 \)
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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

−21.85363225672212651109397425950, −20.818579753642954495742603039419, −19.70626128423457548502374215935, −19.32744211590711981839510651535, −18.57004754655763586456206870239, −18.00213692713837561188782829780, −16.99811153524767777518992095130, −16.29080507917348317679615620753, −15.500168296224425258574185845993, −14.94663972984796856045107047440, −13.67949095680898477545527951560, −13.0537200707658722882783711229, −11.747712185214277579118034801603, −11.071564012124028391173046549038, −10.48708385373338128946355049544, −9.60777950328493976442603143499, −8.80860424084266672783782918891, −7.96082425102150356201505628234, −6.94422962100331485240841253865, −6.32292136951688814468832320022, −5.68955339200828839870569348935, −3.63038338929350607667757040506, −3.23476327561831935042475628595, −2.14114864166251131153169884538, −0.86428014038975039427799977755, 0.28176449617697244172355172159, 1.22877588032906012621540619109, 2.27889735486630212640901509865, 3.41479497303720116504736475412, 4.51226563831613839274348025082, 5.739158609080489343411537767325, 6.50091842828369300856058119836, 7.42276996398123305765552819774, 8.35300771497348958075421502751, 9.06010551208240200251973572514, 9.81092021546129288601878919825, 10.38893848477508471787256078773, 11.65025117917870029352057274294, 12.306100473197306548210163151124, 12.983112818009211521894821033505, 14.00899018450982155611536686521, 15.31198302724204434349386912983, 15.868765076134899893212471435035, 16.586609429209769079073726713563, 17.082874104377481052501187113876, 18.23974939568942291159381705496, 18.63875213907267567643004732435, 19.71014056909487515762423895722, 20.37699262812783016894791617894, 20.706123356414308314187052129772

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