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Properties

Label 1-4563-4563.317-r0-0-0
Degree $1$
Conductor $4563$
Sign $-0.958 + 0.285i$
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  + (−0.682 + 0.730i)2-s + (−0.0670 − 0.997i)4-s + (0.133 − 0.991i)5-s + (0.511 + 0.859i)7-s + (0.774 + 0.632i)8-s + (0.632 + 0.774i)10-s + (0.891 + 0.452i)11-s + (−0.977 − 0.213i)14-s + (−0.991 + 0.133i)16-s + (0.987 + 0.160i)17-s + (−0.866 + 0.5i)19-s + (−0.997 − 0.0670i)20-s + (−0.939 + 0.342i)22-s + (0.173 − 0.984i)23-s + (−0.964 − 0.265i)25-s + ⋯
L(s)  = 1  + (−0.682 + 0.730i)2-s + (−0.0670 − 0.997i)4-s + (0.133 − 0.991i)5-s + (0.511 + 0.859i)7-s + (0.774 + 0.632i)8-s + (0.632 + 0.774i)10-s + (0.891 + 0.452i)11-s + (−0.977 − 0.213i)14-s + (−0.991 + 0.133i)16-s + (0.987 + 0.160i)17-s + (−0.866 + 0.5i)19-s + (−0.997 − 0.0670i)20-s + (−0.939 + 0.342i)22-s + (0.173 − 0.984i)23-s + (−0.964 − 0.265i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.958 + 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $-0.958 + 0.285i$
Analytic conductor: \(21.1904\)
Root analytic conductor: \(21.1904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4563} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4563,\ (0:\ ),\ -0.958 + 0.285i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08685334449 + 0.5958031426i\)
\(L(\frac12)\) \(\approx\) \(0.08685334449 + 0.5958031426i\)
\(L(1)\) \(\approx\) \(0.7098284989 + 0.2374475773i\)
\(L(1)\) \(\approx\) \(0.7098284989 + 0.2374475773i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.682 + 0.730i)T \)
5 \( 1 + (0.133 - 0.991i)T \)
7 \( 1 + (0.511 + 0.859i)T \)
11 \( 1 + (0.891 + 0.452i)T \)
17 \( 1 + (0.987 + 0.160i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (0.173 - 0.984i)T \)
29 \( 1 + (-0.730 - 0.682i)T \)
31 \( 1 + (-0.702 + 0.711i)T \)
37 \( 1 + (-0.903 + 0.428i)T \)
41 \( 1 + (0.989 - 0.147i)T \)
43 \( 1 + (0.252 + 0.967i)T \)
47 \( 1 + (-0.997 - 0.0670i)T \)
53 \( 1 + (0.354 + 0.935i)T \)
59 \( 1 + (-0.133 + 0.991i)T \)
61 \( 1 + (-0.982 - 0.186i)T \)
67 \( 1 + (-0.807 + 0.589i)T \)
71 \( 1 + (0.316 + 0.948i)T \)
73 \( 1 + (-0.391 - 0.919i)T \)
79 \( 1 + (-0.897 + 0.440i)T \)
83 \( 1 + (-0.989 - 0.147i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.133 + 0.991i)T \)
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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

−17.95974274144197309200351292307, −17.3198174260155852237979520083, −16.88562241680011718229795931967, −16.17328913014432177121292564148, −15.13853210248341226842309339972, −14.42684512051436037592752751295, −13.89640351094738216981842661905, −13.1849252727990769111861592737, −12.366266808621226581704829076496, −11.38539088707994229724856868963, −11.19510336482630081214520824921, −10.51526734642834490311758046038, −9.80754052916403918887393953405, −9.17307493486078397481755790690, −8.35909441523708444888544962258, −7.4287331100738996791424068820, −7.20923848405278297665998275574, −6.29089154115503678944708963976, −5.30481528914894118401840969482, −4.148579176802398478501636517795, −3.62858897965319093958112389223, −3.02463527978869162043580449883, −1.91276407329064309379835687312, −1.3990596849498077688465409339, −0.20286091437479733893605496995, 1.29392342316979672449125075619, 1.59900182604475234309812132628, 2.61693739003665843489810264093, 4.09090726714336941159980585034, 4.6219537832556336067815797459, 5.52159870706165954316292320226, 5.9251254705670804775502352395, 6.76072744722903936446308964222, 7.71760668696458716815327314524, 8.28052212529034367197079135272, 8.92089317993585032732552068687, 9.364144346277440168959078580098, 10.13898397820671939252313539420, 10.93501441458280887122830680462, 11.87086422929770772189374518922, 12.3678974361668102305595929823, 13.097738505239398865330523014669, 14.17960707855211831222311231550, 14.62723910710663001731984079084, 15.17230541689660243308385458187, 16.01317817322060275227477871967, 16.67077770390190459809564435536, 17.06087406116287625095275695389, 17.75688086413541222869529164131, 18.40856098651210150165050809778

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