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Properties

Label 1-264-264.29-r0-0-0
Degree $1$
Conductor $264$
Sign $0.530 - 0.847i$
Analytic cond. $1.22601$
Root an. cond. $1.22601$
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  + (0.309 − 0.951i)5-s + (0.809 − 0.587i)7-s + (0.309 + 0.951i)13-s + (0.309 − 0.951i)17-s + (−0.809 − 0.587i)19-s − 23-s + (−0.809 − 0.587i)25-s + (0.809 − 0.587i)29-s + (0.309 + 0.951i)31-s + (−0.309 − 0.951i)35-s + (0.809 − 0.587i)37-s + (−0.809 − 0.587i)41-s + 43-s + (0.809 + 0.587i)47-s + (0.309 − 0.951i)49-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)5-s + (0.809 − 0.587i)7-s + (0.309 + 0.951i)13-s + (0.309 − 0.951i)17-s + (−0.809 − 0.587i)19-s − 23-s + (−0.809 − 0.587i)25-s + (0.809 − 0.587i)29-s + (0.309 + 0.951i)31-s + (−0.309 − 0.951i)35-s + (0.809 − 0.587i)37-s + (−0.809 − 0.587i)41-s + 43-s + (0.809 + 0.587i)47-s + (0.309 − 0.951i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(264\)    =    \(2^{3} \cdot 3 \cdot 11\)
Sign: $0.530 - 0.847i$
Analytic conductor: \(1.22601\)
Root analytic conductor: \(1.22601\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{264} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 264,\ (0:\ ),\ 0.530 - 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.191418341 - 0.6601962288i\)
\(L(\frac12)\) \(\approx\) \(1.191418341 - 0.6601962288i\)
\(L(1)\) \(\approx\) \(1.144035683 - 0.3055820048i\)
\(L(1)\) \(\approx\) \(1.144035683 - 0.3055820048i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 - T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (-0.309 - 0.951i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 - T \)
97 \( 1 + (0.309 + 0.951i)T \)
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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

−25.75485345658697207228994350254, −25.318974807033915202107392505321, −24.132789211601361943138541038565, −23.24015730449709703702087047217, −22.2092726615824456761719956928, −21.53528639541326301939505579243, −20.63405242635465758598571504265, −19.43319783606430810829400569557, −18.44643326100802810767826221991, −17.868250866853831760507195519439, −16.89020176930712590708928811469, −15.42916437367832853065306134835, −14.85577584526771562667713089686, −13.980584494636270912251264541149, −12.786955582877387123001662331489, −11.72368866788856664600021658242, −10.691623304306831091244699209187, −10.0047012009211476699191994792, −8.48026063358359119035452833668, −7.77746810791125818146388337168, −6.31109341506378078587320387054, −5.61624418519599555221190379466, −4.11561736446121772991719454623, −2.82482660767078649826099815198, −1.71322209331769165762374131955, 1.04333786603085578251994304044, 2.26485315094604718881368527777, 4.14781829725371756845274660794, 4.805973135653354682369615252036, 6.07785960333736089882280524007, 7.355058897471709149326810343, 8.438349121488961232824952198690, 9.28756316608019501401830902896, 10.44931489850437159672572061683, 11.565484539753447501328749599900, 12.41944158581162343166729248328, 13.73761943463880195501457237546, 14.11045700231342909242504860218, 15.61201947744269098675482126891, 16.50022096064788967376905955969, 17.32002778449408762628154498815, 18.13449812234842949490341956795, 19.39819407616826526541430765026, 20.337447501773433871439980192699, 21.07072403900069354296569300318, 21.774571352839374117677576130973, 23.27926721696545168431199422288, 23.84287731058887334799650653268, 24.70303566852514171488877484214, 25.59811218562035607463573013475

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