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Properties

Label 1-4563-4563.1024-r0-0-0
Degree $1$
Conductor $4563$
Sign $-0.726 - 0.687i$
Analytic cond. $21.1904$
Root an. cond. $21.1904$
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  + (−0.730 − 0.682i)2-s + (0.0670 + 0.997i)4-s + (−0.611 + 0.791i)5-s + (−0.0134 − 0.999i)7-s + (0.632 − 0.774i)8-s + (0.987 − 0.160i)10-s + (0.452 − 0.891i)11-s + (−0.673 + 0.739i)14-s + (−0.991 + 0.133i)16-s + (−0.354 − 0.935i)17-s + (0.5 − 0.866i)19-s + (−0.830 − 0.556i)20-s + (−0.939 + 0.342i)22-s + (−0.939 + 0.342i)23-s + (−0.252 − 0.967i)25-s + ⋯
L(s)  = 1  + (−0.730 − 0.682i)2-s + (0.0670 + 0.997i)4-s + (−0.611 + 0.791i)5-s + (−0.0134 − 0.999i)7-s + (0.632 − 0.774i)8-s + (0.987 − 0.160i)10-s + (0.452 − 0.891i)11-s + (−0.673 + 0.739i)14-s + (−0.991 + 0.133i)16-s + (−0.354 − 0.935i)17-s + (0.5 − 0.866i)19-s + (−0.830 − 0.556i)20-s + (−0.939 + 0.342i)22-s + (−0.939 + 0.342i)23-s + (−0.252 − 0.967i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $-0.726 - 0.687i$
Analytic conductor: \(21.1904\)
Root analytic conductor: \(21.1904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4563} (1024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4563,\ (0:\ ),\ -0.726 - 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3196570651 - 0.8030132797i\)
\(L(\frac12)\) \(\approx\) \(0.3196570651 - 0.8030132797i\)
\(L(1)\) \(\approx\) \(0.6211154853 - 0.2907649817i\)
\(L(1)\) \(\approx\) \(0.6211154853 - 0.2907649817i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.730 - 0.682i)T \)
5 \( 1 + (-0.611 + 0.791i)T \)
7 \( 1 + (-0.0134 - 0.999i)T \)
11 \( 1 + (0.452 - 0.891i)T \)
17 \( 1 + (-0.354 - 0.935i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.939 + 0.342i)T \)
29 \( 1 + (0.730 + 0.682i)T \)
31 \( 1 + (-0.964 + 0.265i)T \)
37 \( 1 + (0.996 + 0.0804i)T \)
41 \( 1 + (0.783 - 0.621i)T \)
43 \( 1 + (0.964 + 0.265i)T \)
47 \( 1 + (0.897 - 0.440i)T \)
53 \( 1 + (-0.354 - 0.935i)T \)
59 \( 1 + (0.991 + 0.133i)T \)
61 \( 1 + (0.653 - 0.757i)T \)
67 \( 1 + (0.404 - 0.914i)T \)
71 \( 1 + (0.748 + 0.663i)T \)
73 \( 1 + (0.919 - 0.391i)T \)
79 \( 1 + (0.830 + 0.556i)T \)
83 \( 1 + (-0.930 + 0.367i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.379 + 0.925i)T \)
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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

−18.32555553195020155857227146974, −17.81652978152187831528918885448, −17.10158978278568566286101610687, −16.43548174269962455867842959790, −15.840593665653506243321168664, −15.306703849755481228217364904143, −14.6939130328258959392829929725, −14.06703664096150662266909437263, −12.89166513180225014460542057518, −12.390191016352953466437847306103, −11.72284359714268163041851662010, −10.99441085887708456903011380420, −9.995241479985534044004959293348, −9.44400522362404038301928215097, −8.84571352433791551022315664096, −8.09455105381379847422323953144, −7.732886596511137680763214395109, −6.75984933458881927375083148577, −5.890454552733859243209356555402, −5.52762061071040615325427458565, −4.399241042101810730451911951178, −4.047094663835374520630727683588, −2.51937449167786316271626181890, −1.79802793240890371050562561202, −0.961028166018108953891729897198, 0.411942030758156934606003040782, 1.03912079111530798391463271213, 2.302272849722953641125167026839, 2.976913686686306191533176402746, 3.76657803394498538393572388508, 4.15250059107875844452725613876, 5.293880890524220410953935123104, 6.6348096739023903268950770144, 6.94421806814211970534461854326, 7.73221398903423207289729695993, 8.27463183163159569783926151280, 9.265217096923271885890867301624, 9.73682704929924668673670374845, 10.79406853778885926873723047440, 10.984557251147214577146562548030, 11.60428123315163933912495295682, 12.33364047752291494068139141194, 13.24808996334117120306683300696, 13.979495504389436942193714792966, 14.32084813563007719216989142668, 15.60784517840663764463499113804, 16.10836834176450310517626550038, 16.59977437062225645312779358133, 17.60388500697047925662665277554, 17.965694997240712248318041418586

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