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Properties

Label 1-4563-4563.1759-r0-0-0
Degree $1$
Conductor $4563$
Sign $0.122 + 0.992i$
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  + (−0.0670 − 0.997i)2-s + (−0.991 + 0.133i)4-s + (0.252 + 0.967i)5-s + (0.999 − 0.0268i)7-s + (0.200 + 0.979i)8-s + (0.948 − 0.316i)10-s + (0.589 + 0.807i)11-s + (−0.0938 − 0.995i)14-s + (0.964 − 0.265i)16-s + (−0.748 + 0.663i)17-s + (0.5 + 0.866i)19-s + (−0.379 − 0.925i)20-s + (0.766 − 0.642i)22-s + (0.766 − 0.642i)23-s + (−0.872 + 0.488i)25-s + ⋯
L(s)  = 1  + (−0.0670 − 0.997i)2-s + (−0.991 + 0.133i)4-s + (0.252 + 0.967i)5-s + (0.999 − 0.0268i)7-s + (0.200 + 0.979i)8-s + (0.948 − 0.316i)10-s + (0.589 + 0.807i)11-s + (−0.0938 − 0.995i)14-s + (0.964 − 0.265i)16-s + (−0.748 + 0.663i)17-s + (0.5 + 0.866i)19-s + (−0.379 − 0.925i)20-s + (0.766 − 0.642i)22-s + (0.766 − 0.642i)23-s + (−0.872 + 0.488i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4563 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.122 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4563\)    =    \(3^{3} \cdot 13^{2}\)
Sign: $0.122 + 0.992i$
Analytic conductor: \(21.1904\)
Root analytic conductor: \(21.1904\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4563} (1759, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4563,\ (0:\ ),\ 0.122 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.062910648 + 0.9400251943i\)
\(L(\frac12)\) \(\approx\) \(1.062910648 + 0.9400251943i\)
\(L(1)\) \(\approx\) \(1.044326793 - 0.03650792398i\)
\(L(1)\) \(\approx\) \(1.044326793 - 0.03650792398i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.0670 - 0.997i)T \)
5 \( 1 + (0.252 + 0.967i)T \)
7 \( 1 + (0.999 - 0.0268i)T \)
11 \( 1 + (0.589 + 0.807i)T \)
17 \( 1 + (-0.748 + 0.663i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.766 - 0.642i)T \)
29 \( 1 + (0.0670 + 0.997i)T \)
31 \( 1 + (-0.859 + 0.511i)T \)
37 \( 1 + (-0.987 - 0.160i)T \)
41 \( 1 + (-0.226 + 0.974i)T \)
43 \( 1 + (0.859 + 0.511i)T \)
47 \( 1 + (-0.611 + 0.791i)T \)
53 \( 1 + (-0.748 + 0.663i)T \)
59 \( 1 + (-0.964 - 0.265i)T \)
61 \( 1 + (-0.147 - 0.989i)T \)
67 \( 1 + (0.673 + 0.739i)T \)
71 \( 1 + (-0.120 - 0.992i)T \)
73 \( 1 + (-0.692 + 0.721i)T \)
79 \( 1 + (0.379 + 0.925i)T \)
83 \( 1 + (-0.730 + 0.682i)T \)
89 \( 1 - T \)
97 \( 1 + (0.711 + 0.702i)T \)
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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

−17.81564584078817818910753692021, −17.170147829898931095169013883388, −16.985844642450476132832700296649, −15.90029153790045843329895042419, −15.63510169301018250271223527493, −14.753632138561949758636776739581, −13.96686381362285255701045680744, −13.56217052603502686527078447298, −12.99256568441626023066561187400, −11.907815257092514513616414838776, −11.43202656768640604115419817492, −10.513863328550764669701761555872, −9.43635421009796976411745440464, −8.97970693876399641860748802488, −8.573079863342125687979048016404, −7.67479037067930076183912236910, −7.11605372438587601987151269857, −6.16840350221605811846252798717, −5.42351591567652836070730664158, −4.95882565366845509479741436483, −4.27668601611879778785967743584, −3.44257032714278174077133488186, −2.11084113388283826974543555420, −1.19121856848137607108421514028, −0.40613961942027414048844418798, 1.51365039906593960610294120223, 1.62367982440840659150725808684, 2.67579589610098991405343885287, 3.40084893049588318615029130856, 4.23017304178348517714321876664, 4.85619902524581534929337619859, 5.70669113710465861248095207370, 6.66954691607615718837864506518, 7.4051175939316486922656755542, 8.183716056359878768385352443905, 8.942586299055592744350223516069, 9.65040495928698229396493686148, 10.362207454043726540360365185286, 11.10276260034504349028917785566, 11.22863738660794558791223804832, 12.43367167983038460501234867951, 12.624753961807539024673777546470, 13.81293384707555279488045978433, 14.33655901806428644091714376802, 14.694168876041371549138864834932, 15.44411809101742883009290205822, 16.666126682475250734543725794880, 17.380959655193407314418106610025, 17.864452774609880422731926699227, 18.35088642195014519823313870955

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