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Properties

Label 1-4563-4563.1291-r0-0-0
Degree $1$
Conductor $4563$
Sign $0.865 + 0.501i$
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  + (0.452 + 0.891i)2-s + (−0.589 + 0.807i)4-s + (−0.977 + 0.213i)5-s + (0.982 − 0.186i)7-s + (−0.987 − 0.160i)8-s + (−0.632 − 0.774i)10-s + (0.956 + 0.291i)11-s + (0.611 + 0.791i)14-s + (−0.303 − 0.952i)16-s + (−0.354 − 0.935i)17-s + (0.5 + 0.866i)19-s + (0.404 − 0.914i)20-s + (0.173 + 0.984i)22-s + (0.173 + 0.984i)23-s + (0.909 − 0.416i)25-s + ⋯
L(s)  = 1  + (0.452 + 0.891i)2-s + (−0.589 + 0.807i)4-s + (−0.977 + 0.213i)5-s + (0.982 − 0.186i)7-s + (−0.987 − 0.160i)8-s + (−0.632 − 0.774i)10-s + (0.956 + 0.291i)11-s + (0.611 + 0.791i)14-s + (−0.303 − 0.952i)16-s + (−0.354 − 0.935i)17-s + (0.5 + 0.866i)19-s + (0.404 − 0.914i)20-s + (0.173 + 0.984i)22-s + (0.173 + 0.984i)23-s + (0.909 − 0.416i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $0.865 + 0.501i$
Analytic conductor: \(21.1904\)
Root analytic conductor: \(21.1904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4563} (1291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4563,\ (0:\ ),\ 0.865 + 0.501i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.744919616 + 0.4688376585i\)
\(L(\frac12)\) \(\approx\) \(1.744919616 + 0.4688376585i\)
\(L(1)\) \(\approx\) \(1.085703542 + 0.5166195930i\)
\(L(1)\) \(\approx\) \(1.085703542 + 0.5166195930i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.452 + 0.891i)T \)
5 \( 1 + (-0.977 + 0.213i)T \)
7 \( 1 + (0.982 - 0.186i)T \)
11 \( 1 + (0.956 + 0.291i)T \)
17 \( 1 + (-0.354 - 0.935i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.173 + 0.984i)T \)
29 \( 1 + (-0.452 - 0.891i)T \)
31 \( 1 + (0.815 - 0.579i)T \)
37 \( 1 + (-0.428 - 0.903i)T \)
41 \( 1 + (0.999 + 0.0268i)T \)
43 \( 1 + (-0.815 - 0.579i)T \)
47 \( 1 + (-0.994 + 0.107i)T \)
53 \( 1 + (-0.354 - 0.935i)T \)
59 \( 1 + (0.303 - 0.952i)T \)
61 \( 1 + (0.859 + 0.511i)T \)
67 \( 1 + (0.897 - 0.440i)T \)
71 \( 1 + (0.748 + 0.663i)T \)
73 \( 1 + (-0.799 - 0.600i)T \)
79 \( 1 + (-0.404 + 0.914i)T \)
83 \( 1 + (-0.523 - 0.852i)T \)
89 \( 1 - T \)
97 \( 1 + (0.673 - 0.739i)T \)
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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

−18.36203718806357091209967147793, −17.60228278399814994069913992245, −16.96346727074353842298037952854, −16.009133245988679339000990683535, −15.257162488161307352937430366739, −14.680962047413082090160222948379, −14.23845270347070648804751964519, −13.32842938692031071111024229337, −12.594668995081603257123463166629, −12.00993653442077860021968672637, −11.35791839310075662674001269085, −11.03554322795534851950287644660, −10.22461349885858467601647796247, −9.17751444151058656253207816332, −8.615372555326201113873396686841, −8.15062372138688601874210350446, −6.98997476649162505616670789122, −6.31811715805289116018226161923, −5.2416285070167843593974713366, −4.67315865904017593756436783431, −4.10445233278201789787508843509, −3.33654844512963238867447374155, −2.54813879362759556151895285496, −1.45069463495986789405164911398, −0.967576450972296230893750524690, 0.51387966098326908448645700126, 1.70766259152343952723485066210, 2.86229988177764325871203168593, 3.84373342544555345218258559396, 4.13152348906430986117833096209, 5.00442467282395122887338174979, 5.638255934455268083916699492717, 6.70754436839399605701483167826, 7.177912024340470468443015837011, 7.90726986681743650300258322000, 8.29987277131642098487898875921, 9.262664087887075843213168232618, 9.883449460895769400052374755529, 11.22996935188519022051407604548, 11.58047156390441908279505454453, 12.08824916505707463432947620662, 12.996120291973441727284669811362, 13.841855442993454180752671220144, 14.398451123102149020067528287862, 14.85206594875706267596954795805, 15.624139777305822439741140646837, 16.072597051269406745693830011427, 16.93475864734880330561123486433, 17.472345105237974558181539834088, 18.10466854568888301514862339220

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