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Properties

Label 1-4563-4563.1258-r0-0-0
Degree $1$
Conductor $4563$
Sign $0.421 + 0.906i$
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  + (0.673 + 0.739i)2-s + (−0.0938 + 0.995i)4-s + (−0.329 + 0.944i)5-s + (0.956 − 0.291i)7-s + (−0.799 + 0.600i)8-s + (−0.919 + 0.391i)10-s + (−0.611 − 0.791i)11-s + (0.859 + 0.511i)14-s + (−0.982 − 0.186i)16-s + (0.120 + 0.992i)17-s + (0.5 − 0.866i)19-s + (−0.909 − 0.416i)20-s + (0.173 − 0.984i)22-s + (0.173 − 0.984i)23-s + (−0.783 − 0.621i)25-s + ⋯
L(s)  = 1  + (0.673 + 0.739i)2-s + (−0.0938 + 0.995i)4-s + (−0.329 + 0.944i)5-s + (0.956 − 0.291i)7-s + (−0.799 + 0.600i)8-s + (−0.919 + 0.391i)10-s + (−0.611 − 0.791i)11-s + (0.859 + 0.511i)14-s + (−0.982 − 0.186i)16-s + (0.120 + 0.992i)17-s + (0.5 − 0.866i)19-s + (−0.909 − 0.416i)20-s + (0.173 − 0.984i)22-s + (0.173 − 0.984i)23-s + (−0.783 − 0.621i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.421 + 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $0.421 + 0.906i$
Analytic conductor: \(21.1904\)
Root analytic conductor: \(21.1904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4563} (1258, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4563,\ (0:\ ),\ 0.421 + 0.906i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.030646570 + 1.295385022i\)
\(L(\frac12)\) \(\approx\) \(2.030646570 + 1.295385022i\)
\(L(1)\) \(\approx\) \(1.286862606 + 0.7128699671i\)
\(L(1)\) \(\approx\) \(1.286862606 + 0.7128699671i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.673 + 0.739i)T \)
5 \( 1 + (-0.329 + 0.944i)T \)
7 \( 1 + (0.956 - 0.291i)T \)
11 \( 1 + (-0.611 - 0.791i)T \)
17 \( 1 + (0.120 + 0.992i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.673 - 0.739i)T \)
31 \( 1 + (-0.930 - 0.367i)T \)
37 \( 1 + (0.200 - 0.979i)T \)
41 \( 1 + (0.589 + 0.807i)T \)
43 \( 1 + (0.930 - 0.367i)T \)
47 \( 1 + (0.815 - 0.579i)T \)
53 \( 1 + (0.120 + 0.992i)T \)
59 \( 1 + (0.982 - 0.186i)T \)
61 \( 1 + (0.998 - 0.0536i)T \)
67 \( 1 + (-0.964 + 0.265i)T \)
71 \( 1 + (0.970 + 0.239i)T \)
73 \( 1 + (0.845 + 0.534i)T \)
79 \( 1 + (0.909 + 0.416i)T \)
83 \( 1 + (0.404 + 0.914i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.653 + 0.757i)T \)
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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

−18.07623622693376237440256441303, −17.74278301719443188426562075423, −16.63774844784905907547602724688, −15.92985443008164157981783695925, −15.339538100210679620274691540752, −14.650022666522212855112506255962, −13.9968696969476348279333440059, −13.28469880590411384889181406893, −12.530227961611389095985976333117, −12.1421438975142709212015149313, −11.42033233072077965632576240581, −10.896203282251013321933935264234, −9.890853667043880197234452594050, −9.35288030281380828663566492346, −8.64591509197897078648884373652, −7.66254111593726509547022814794, −7.224449772874176327332129192522, −5.79982655909127450861966722740, −5.25808621995329222777556249994, −4.87294462121727248456947837550, −4.0489705769083819127948323446, −3.29907093700564841057482402736, −2.239850820072382642798954158640, −1.627070070747681348687592322891, −0.84076511996611327904246085891, 0.645580532640542980388247845664, 2.247543081652342037024345042928, 2.70427560899605425120169550395, 3.87156011792112163216155383140, 4.073961701642606491628381678555, 5.24074261443130647619436915547, 5.7393778840688984738231858145, 6.5635100384280003416510381426, 7.33132249501410024750360283031, 7.83022059518251770206883592123, 8.41356104568199517689517243220, 9.25248221976942171456090150305, 10.455838730632308669840020635048, 11.09319375805611446719028088173, 11.39986495552437873544784008190, 12.41835178572537885377503800309, 13.09849678479100670275404549857, 13.88752926568993073443857010601, 14.312813391754803891748921458900, 15.04939566140333900746862994836, 15.395541980015904569910696977912, 16.291998230480029878855272365483, 16.88037171356611321870881269757, 17.68144372327298549848786414689, 18.182825392640606480322310798580

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