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Properties

Label 1-1012-1012.295-r0-0-0
Degree $1$
Conductor $1012$
Sign $-0.612 + 0.790i$
Analytic cond. $4.69970$
Root an. cond. $4.69970$
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  + (0.254 + 0.967i)3-s + (0.870 + 0.491i)5-s + (−0.564 − 0.825i)7-s + (−0.870 + 0.491i)9-s + (0.610 + 0.791i)13-s + (−0.254 + 0.967i)15-s + (0.466 + 0.884i)17-s + (−0.985 − 0.170i)19-s + (0.654 − 0.755i)21-s + (0.516 + 0.856i)25-s + (−0.696 − 0.717i)27-s + (−0.985 + 0.170i)29-s + (0.362 + 0.931i)31-s + (−0.0855 − 0.996i)35-s + (−0.993 − 0.113i)37-s + ⋯
L(s)  = 1  + (0.254 + 0.967i)3-s + (0.870 + 0.491i)5-s + (−0.564 − 0.825i)7-s + (−0.870 + 0.491i)9-s + (0.610 + 0.791i)13-s + (−0.254 + 0.967i)15-s + (0.466 + 0.884i)17-s + (−0.985 − 0.170i)19-s + (0.654 − 0.755i)21-s + (0.516 + 0.856i)25-s + (−0.696 − 0.717i)27-s + (−0.985 + 0.170i)29-s + (0.362 + 0.931i)31-s + (−0.0855 − 0.996i)35-s + (−0.993 − 0.113i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1012 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1012\)    =    \(2^{2} \cdot 11 \cdot 23\)
Sign: $-0.612 + 0.790i$
Analytic conductor: \(4.69970\)
Root analytic conductor: \(4.69970\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1012} (295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1012,\ (0:\ ),\ -0.612 + 0.790i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6580975250 + 1.341758994i\)
\(L(\frac12)\) \(\approx\) \(0.6580975250 + 1.341758994i\)
\(L(1)\) \(\approx\) \(1.027683412 + 0.5639986449i\)
\(L(1)\) \(\approx\) \(1.027683412 + 0.5639986449i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.254 + 0.967i)T \)
5 \( 1 + (0.870 + 0.491i)T \)
7 \( 1 + (-0.564 - 0.825i)T \)
13 \( 1 + (0.610 + 0.791i)T \)
17 \( 1 + (0.466 + 0.884i)T \)
19 \( 1 + (-0.985 - 0.170i)T \)
29 \( 1 + (-0.985 + 0.170i)T \)
31 \( 1 + (0.362 + 0.931i)T \)
37 \( 1 + (-0.993 - 0.113i)T \)
41 \( 1 + (0.993 - 0.113i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.309 + 0.951i)T \)
53 \( 1 + (-0.941 - 0.336i)T \)
59 \( 1 + (0.0285 + 0.999i)T \)
61 \( 1 + (0.998 - 0.0570i)T \)
67 \( 1 + (-0.654 + 0.755i)T \)
71 \( 1 + (-0.0855 + 0.996i)T \)
73 \( 1 + (0.897 - 0.441i)T \)
79 \( 1 + (0.610 + 0.791i)T \)
83 \( 1 + (0.198 + 0.980i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (-0.198 + 0.980i)T \)
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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

−21.05744484318259099479554512293, −20.71051651003780953913567844980, −19.70452713642121856601734971888, −18.86069461774212412132090216454, −18.32907993169592448975572404665, −17.55776701833199547743240678654, −16.80739632005517933353591723762, −15.864133256379892567996753290143, −14.92983423738011200846608044734, −14.06855102783934623956940627784, −13.21007731480741567691210953161, −12.78852397401593514537392141513, −12.05654977257712788161996970332, −11.03599660000650444470487151100, −9.849385815886348992292675300615, −9.135796878261504938067324044951, −8.42554010974784257853763983465, −7.55185047913758509302517805624, −6.33184153380505978244277553234, −5.93781747170961982320523917609, −5.060718523498311241263468568906, −3.50095328292325561654812034224, −2.55610091086238042718282870236, −1.82475487143279641469775343622, −0.5915507470770368409727538017, 1.51350457960467284486319114336, 2.63044757109279040657620329824, 3.64177698363551228515931293910, 4.22945958276521289576964079249, 5.46510710990197018749604503751, 6.27284556178896488418340035444, 7.07100509141381560553179833859, 8.326407319343232392217614521091, 9.19876103628340393857284809454, 9.851246288568058226653065128298, 10.74361149085735134008699881573, 10.968219289994300349873857736917, 12.50502532468329673966167549292, 13.39811140488026717431418234476, 14.13221703418422409701290929184, 14.65129088515574811128863840801, 15.6609873053625765958758150754, 16.47033001171370780657657067604, 17.08776741940300822139552411467, 17.7676015105159427926765285044, 19.147211865732538035998702681746, 19.36667683827111447711797097262, 20.734682844645831818144162817566, 21.00067004440836727765716950908, 21.84705814325985164088355885840

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