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Properties

Label 1-4760-4760.1549-r0-0-0
Degree $1$
Conductor $4760$
Sign $0.275 - 0.961i$
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  + (0.258 + 0.965i)3-s + (−0.866 + 0.5i)9-s + (0.965 − 0.258i)11-s − 13-s + (−0.866 + 0.5i)19-s + (−0.258 + 0.965i)23-s + (−0.707 − 0.707i)27-s + (0.707 − 0.707i)29-s + (0.258 + 0.965i)31-s + (0.5 + 0.866i)33-s + (−0.965 − 0.258i)37-s + (−0.258 − 0.965i)39-s + (−0.707 − 0.707i)41-s − i·43-s + (−0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)3-s + (−0.866 + 0.5i)9-s + (0.965 − 0.258i)11-s − 13-s + (−0.866 + 0.5i)19-s + (−0.258 + 0.965i)23-s + (−0.707 − 0.707i)27-s + (0.707 − 0.707i)29-s + (0.258 + 0.965i)31-s + (0.5 + 0.866i)33-s + (−0.965 − 0.258i)37-s + (−0.258 − 0.965i)39-s + (−0.707 − 0.707i)41-s − i·43-s + (−0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4760 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.275 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4760\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 17\)
Sign: $0.275 - 0.961i$
Analytic conductor: \(22.1053\)
Root analytic conductor: \(22.1053\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4760} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4760,\ (0:\ ),\ 0.275 - 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4794341845 - 0.3615082771i\)
\(L(\frac12)\) \(\approx\) \(0.4794341845 - 0.3615082771i\)
\(L(1)\) \(\approx\) \(0.8964900106 + 0.2480937604i\)
\(L(1)\) \(\approx\) \(0.8964900106 + 0.2480937604i\)

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 \( 1 + (0.258 + 0.965i)T \)
11 \( 1 + (0.965 - 0.258i)T \)
13 \( 1 - T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.258 + 0.965i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (0.258 + 0.965i)T \)
37 \( 1 + (-0.965 - 0.258i)T \)
41 \( 1 + (-0.707 - 0.707i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.258 - 0.965i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.258 + 0.965i)T \)
79 \( 1 + (0.258 - 0.965i)T \)
83 \( 1 - iT \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.707 - 0.707i)T \)
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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

−18.34000551502316252053476355438, −17.55101146219225923617404176072, −17.116904569011184811540102675093, −16.51907911292878978404145406067, −15.43033432939846999131103792980, −14.68773420147223021103357600528, −14.38149194966789611984395259700, −13.57144362001983682094891478519, −12.81918366128966383003779010436, −12.27919168378692350007144726494, −11.7603898619203591919383887421, −10.95568355660344641692819311259, −10.074400140320878128738023772086, −9.28181885167279276750446287959, −8.68791578712790908059127350298, −7.94565698148229295085380709996, −7.26373323066626594317468346270, −6.45979679808130301518312251391, −6.23038711680064483605096147080, −4.92881925202337439052731393420, −4.390678470376935549326110635074, −3.28485365786065286076072078073, −2.57511562704275116838490302233, −1.84158676613036967415542630382, −1.018241846526145928854834302857, 0.156650588727731560560388166026, 1.644708044237540223468223834713, 2.3693671016130372280575564614, 3.44880074583953836323492337942, 3.80087792812307916888651586607, 4.76685079988833927629852722809, 5.27744304663824431952574879654, 6.21597702210377349023229117892, 6.91321907705601700786350188933, 7.88396187140543288158855319429, 8.57098277259237611072726355106, 9.159011398170656727244508747492, 9.933186893576065583847815507015, 10.35213059681203390411363953625, 11.20665764287824267502607902905, 11.93130090389543097177564847130, 12.42012651250208350433708017125, 13.586329981356605776276112436263, 14.134288166285138509496312988869, 14.63286576431102683828721872143, 15.42357244523053107452552065837, 15.84807753213833136160306995949, 16.825430628796114475285300112242, 17.14029712712123256856132876750, 17.73573558097533000816842936342

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