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Properties

Label 1-4563-4563.1777-r0-0-0
Degree $1$
Conductor $4563$
Sign $0.0261 - 0.999i$
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  + (0.476 − 0.879i)2-s + (−0.545 − 0.837i)4-s + (−0.589 + 0.807i)5-s + (0.909 − 0.416i)7-s + (−0.996 + 0.0804i)8-s + (0.428 + 0.903i)10-s + (−0.783 + 0.621i)11-s + (0.0670 − 0.997i)14-s + (−0.404 + 0.914i)16-s + (0.568 + 0.822i)17-s + (−0.5 − 0.866i)19-s + (0.998 + 0.0536i)20-s + (0.173 + 0.984i)22-s + (0.173 + 0.984i)23-s + (−0.303 − 0.952i)25-s + ⋯
L(s)  = 1  + (0.476 − 0.879i)2-s + (−0.545 − 0.837i)4-s + (−0.589 + 0.807i)5-s + (0.909 − 0.416i)7-s + (−0.996 + 0.0804i)8-s + (0.428 + 0.903i)10-s + (−0.783 + 0.621i)11-s + (0.0670 − 0.997i)14-s + (−0.404 + 0.914i)16-s + (0.568 + 0.822i)17-s + (−0.5 − 0.866i)19-s + (0.998 + 0.0536i)20-s + (0.173 + 0.984i)22-s + (0.173 + 0.984i)23-s + (−0.303 − 0.952i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0261 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0261 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $0.0261 - 0.999i$
Analytic conductor: \(21.1904\)
Root analytic conductor: \(21.1904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4563} (1777, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4563,\ (0:\ ),\ 0.0261 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.118663672 - 1.089772666i\)
\(L(\frac12)\) \(\approx\) \(1.118663672 - 1.089772666i\)
\(L(1)\) \(\approx\) \(1.015683621 - 0.4629052125i\)
\(L(1)\) \(\approx\) \(1.015683621 - 0.4629052125i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.476 - 0.879i)T \)
5 \( 1 + (-0.589 + 0.807i)T \)
7 \( 1 + (0.909 - 0.416i)T \)
11 \( 1 + (-0.783 + 0.621i)T \)
17 \( 1 + (0.568 + 0.822i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (0.476 - 0.879i)T \)
31 \( 1 + (-0.673 - 0.739i)T \)
37 \( 1 + (-0.845 + 0.534i)T \)
41 \( 1 + (-0.872 - 0.488i)T \)
43 \( 1 + (-0.673 + 0.739i)T \)
47 \( 1 + (-0.452 - 0.891i)T \)
53 \( 1 + (0.568 + 0.822i)T \)
59 \( 1 + (-0.404 - 0.914i)T \)
61 \( 1 + (-0.711 - 0.702i)T \)
67 \( 1 + (0.730 + 0.682i)T \)
71 \( 1 + (-0.354 - 0.935i)T \)
73 \( 1 + (0.948 - 0.316i)T \)
79 \( 1 + (0.998 + 0.0536i)T \)
83 \( 1 + (0.859 + 0.511i)T \)
89 \( 1 + T \)
97 \( 1 + (0.994 - 0.107i)T \)
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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

−18.46155990754095115762129863487, −17.604799882893517448201219262, −16.75650559447483465706473403033, −16.32589250457916471300407828739, −15.81468948521932637016671603675, −15.01470140033239006979021645215, −14.459268924094017272400669385537, −13.81718903449492285967537174628, −13.017131053462478591781692132224, −12.29779580705117646382425896238, −11.97747887793324272892922577762, −11.05313580048766732973472737376, −10.24271626292078422079641357087, −9.013051328264768363262586965533, −8.62814172120421553079351180047, −8.03982213274484951740734235092, −7.48517160753853820701406538988, −6.60341053684273791049153086196, −5.59683305993986290603048778895, −5.11915424335947942518767824909, −4.67058597600092725333633171619, −3.6782748539829199225197557390, −3.02162479677236440556589150324, −1.887634673461018324758624607921, −0.717337216482545481824119192651, 0.502388135583438286110853897524, 1.75567905877179840325743618985, 2.234171780847512481059524168556, 3.275854638442049453992090651257, 3.794266216644080001016537404545, 4.67989701752243664046462542236, 5.15129837632832846070369780992, 6.16097016358999299171001493345, 6.977193129863455890096728768828, 7.8146985994440792895828809196, 8.31657588397764757090591039451, 9.39673030229869561079103607261, 10.26026342470967765514125287689, 10.60220974871212792671101407153, 11.32159633543658165413596646084, 11.82399415901222364785499351066, 12.56324687850258388595556605684, 13.43341819915756123378874954805, 13.84994691869364612233163872324, 14.828042334467777241843165019863, 15.12672718906172452484623431439, 15.622029660442446929981404154669, 16.96946045271024327011215289139, 17.56854437915830027621027001717, 18.239516108991388231908902546

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