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Properties

Label 1-4760-4760.1213-r0-0-0
Degree $1$
Conductor $4760$
Sign $0.932 + 0.362i$
Analytic cond. $22.1053$
Root an. cond. $22.1053$
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  + (0.991 + 0.130i)3-s + (0.965 + 0.258i)9-s + (−0.793 + 0.608i)11-s − 13-s + (0.258 − 0.965i)19-s + (0.991 − 0.130i)23-s + (0.923 + 0.382i)27-s + (0.382 + 0.923i)29-s + (0.130 − 0.991i)31-s + (−0.866 + 0.5i)33-s + (0.608 − 0.793i)37-s + (−0.991 − 0.130i)39-s + (−0.382 + 0.923i)41-s + (−0.707 − 0.707i)43-s + (0.5 + 0.866i)47-s + ⋯
L(s)  = 1  + (0.991 + 0.130i)3-s + (0.965 + 0.258i)9-s + (−0.793 + 0.608i)11-s − 13-s + (0.258 − 0.965i)19-s + (0.991 − 0.130i)23-s + (0.923 + 0.382i)27-s + (0.382 + 0.923i)29-s + (0.130 − 0.991i)31-s + (−0.866 + 0.5i)33-s + (0.608 − 0.793i)37-s + (−0.991 − 0.130i)39-s + (−0.382 + 0.923i)41-s + (−0.707 − 0.707i)43-s + (0.5 + 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.932 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4760\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.932 + 0.362i$
Analytic conductor: \(22.1053\)
Root analytic conductor: \(22.1053\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4760} (1213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4760,\ (0:\ ),\ 0.932 + 0.362i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.452849252 + 0.4599920375i\)
\(L(\frac12)\) \(\approx\) \(2.452849252 + 0.4599920375i\)
\(L(1)\) \(\approx\) \(1.452662205 + 0.1184510198i\)
\(L(1)\) \(\approx\) \(1.452662205 + 0.1184510198i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + (0.991 + 0.130i)T \)
11 \( 1 + (-0.793 + 0.608i)T \)
13 \( 1 - T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (0.991 - 0.130i)T \)
29 \( 1 + (0.382 + 0.923i)T \)
31 \( 1 + (0.130 - 0.991i)T \)
37 \( 1 + (0.608 - 0.793i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.258 + 0.965i)T \)
59 \( 1 + (-0.258 - 0.965i)T \)
61 \( 1 + (0.991 - 0.130i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.923 - 0.382i)T \)
73 \( 1 + (-0.130 + 0.991i)T \)
79 \( 1 + (-0.130 - 0.991i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + (0.923 - 0.382i)T \)
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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

−18.480322645104661563865649559427, −17.42558250576645954131746585121, −16.750318021176328086039733914954, −15.97333065453123283349358084713, −15.347603075895776856503445794429, −14.73400968633945215352075146729, −14.0940608953306652459804127180, −13.458603138761671332391331536774, −12.87010618332117353901029805867, −12.15564802542065673061352978759, −11.43178508527613665854515163445, −10.31792662664129540858191590891, −10.043532665452856249256311377540, −9.18907546566939937035835153420, −8.398548885925672275812254428051, −7.961043913440346053910134124403, −7.196957330042615590375189533864, −6.54939043096766341740150844828, −5.46691883819850232106877469228, −4.85957335799465827463626852729, −3.914719330221394494403728779056, −3.137959994111002633869285602221, −2.58216618541242290739088320070, −1.74953059088722076494668746551, −0.73155390188504033287023079528, 0.8184553547233068378390094020, 1.97622172363282374916947616083, 2.62336648069743452988552158066, 3.137820396399598967623285461171, 4.24722914786307708574447361675, 4.812002733924280010355725597386, 5.46139757653809688542431492496, 6.8036601347237436460580680820, 7.20777659392701308571260754421, 7.90695996481630064904432072986, 8.607998117733742472799160428504, 9.45913291345682838725064520748, 9.78914231021544918022142606340, 10.64842235084252926259290987444, 11.320961942536675889519648928215, 12.41434230718581953271756693699, 12.84764048907047013281006200921, 13.48316996515708481721640510315, 14.25881508373508337657539879034, 14.89436543256474247587934399520, 15.37615096022317200453900098877, 15.98683871783482370149616057304, 16.86198691363240984189856211895, 17.54293361902665073590270379024, 18.32593595859962457816303480684

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