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Properties

Label 1-1021-1021.392-r1-0-0
Degree $1$
Conductor $1021$
Sign $0.835 + 0.549i$
Analytic cond. $109.721$
Root an. cond. $109.721$
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  + (0.361 − 0.932i)2-s + (0.850 + 0.526i)3-s + (−0.739 − 0.673i)4-s + (−0.602 − 0.798i)5-s + (0.798 − 0.602i)6-s + (0.183 − 0.982i)7-s + (−0.895 + 0.445i)8-s + (0.445 + 0.895i)9-s + (−0.961 + 0.273i)10-s + (−0.273 + 0.961i)11-s + (−0.273 − 0.961i)12-s + (−0.361 − 0.932i)13-s + (−0.850 − 0.526i)14-s + (−0.0922 − 0.995i)15-s + (0.0922 + 0.995i)16-s + (0.602 + 0.798i)17-s + ⋯
L(s)  = 1  + (0.361 − 0.932i)2-s + (0.850 + 0.526i)3-s + (−0.739 − 0.673i)4-s + (−0.602 − 0.798i)5-s + (0.798 − 0.602i)6-s + (0.183 − 0.982i)7-s + (−0.895 + 0.445i)8-s + (0.445 + 0.895i)9-s + (−0.961 + 0.273i)10-s + (−0.273 + 0.961i)11-s + (−0.273 − 0.961i)12-s + (−0.361 − 0.932i)13-s + (−0.850 − 0.526i)14-s + (−0.0922 − 0.995i)15-s + (0.0922 + 0.995i)16-s + (0.602 + 0.798i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1021\)
Sign: $0.835 + 0.549i$
Analytic conductor: \(109.721\)
Root analytic conductor: \(109.721\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1021} (392, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1021,\ (1:\ ),\ 0.835 + 0.549i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.118017309 + 0.3350382827i\)
\(L(\frac12)\) \(\approx\) \(1.118017309 + 0.3350382827i\)
\(L(1)\) \(\approx\) \(1.058236107 - 0.5216133803i\)
\(L(1)\) \(\approx\) \(1.058236107 - 0.5216133803i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1021 \( 1 \)
good2 \( 1 + (0.361 - 0.932i)T \)
3 \( 1 + (0.850 + 0.526i)T \)
5 \( 1 + (-0.602 - 0.798i)T \)
7 \( 1 + (0.183 - 0.982i)T \)
11 \( 1 + (-0.273 + 0.961i)T \)
13 \( 1 + (-0.361 - 0.932i)T \)
17 \( 1 + (0.602 + 0.798i)T \)
19 \( 1 + (-0.895 - 0.445i)T \)
23 \( 1 + (0.0922 - 0.995i)T \)
29 \( 1 + (0.982 - 0.183i)T \)
31 \( 1 + (-0.895 + 0.445i)T \)
37 \( 1 + (-0.526 + 0.850i)T \)
41 \( 1 + (-0.445 + 0.895i)T \)
43 \( 1 + (0.183 - 0.982i)T \)
47 \( 1 + (-0.850 - 0.526i)T \)
53 \( 1 + (-0.995 - 0.0922i)T \)
59 \( 1 + (-0.526 + 0.850i)T \)
61 \( 1 + (0.850 + 0.526i)T \)
67 \( 1 + T \)
71 \( 1 + (0.445 + 0.895i)T \)
73 \( 1 + (-0.850 + 0.526i)T \)
79 \( 1 + (0.273 + 0.961i)T \)
83 \( 1 + (0.850 + 0.526i)T \)
89 \( 1 + (0.739 - 0.673i)T \)
97 \( 1 + (-0.183 + 0.982i)T \)
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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

−21.51710664505375309207455181700, −20.80352455356740461653330557441, −19.22680285657559770181814700579, −19.06369228655800392148551011958, −18.32430917309641968842199396007, −17.56442253876632780690084600637, −16.251228479843441719595083526278, −15.75343284717850269968199207759, −14.85937943643836246248485592552, −14.33775340805804623671508322969, −13.7961209883669334828898372883, −12.674314781973043927251698121535, −12.02605762339392913462202218421, −11.19401851414615573245804910507, −9.66345047602898259364419202446, −8.93408717837352043020110700641, −8.126725935826186434239022446275, −7.55143320930216638022902616347, −6.65325398523285908082664170469, −5.958623928202785863971761818546, −4.81396177520142556293907398081, −3.61617364308187321356297296479, −3.062534180869551601007188812584, −2.01537711728182971616044996303, −0.204149420480593163865083485490, 1.01406866866538839127349017151, 2.02975598023875345083367109265, 3.13639745796095530173299597587, 3.97440122978851318303571229449, 4.63421869641961602739726235987, 5.20807055099291783011439893187, 6.91291478755572599936451390506, 8.10785087034172630264855229264, 8.45823575835859563477016724151, 9.65155036981467461475212058464, 10.31746275630852261560719940605, 10.82205851107903151246003888052, 12.11808559681810774585741021099, 12.84780002569841478325692974653, 13.295816975332779434075287653207, 14.45938723943374697538880189615, 14.94114415444519459974375436687, 15.72795649554856863539657089647, 16.86040955300790760712933356554, 17.53693379338718618145503250214, 18.73620308332906084396816652147, 19.64194336912432240247931047057, 20.01175754734439828644812658796, 20.59239341757436082363661403644, 21.16379120784303479184281428796

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