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Properties

Label 1-6015-6015.3833-r0-0-0
Degree $1$
Conductor $6015$
Sign $0.283 + 0.958i$
Analytic cond. $27.9335$
Root an. cond. $27.9335$
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  + (−0.125 − 0.992i)2-s + (−0.968 + 0.248i)4-s + (−0.998 − 0.0627i)7-s + (0.368 + 0.929i)8-s + (−0.728 − 0.684i)11-s + (−0.684 + 0.728i)13-s + (0.0627 + 0.998i)14-s + (0.876 − 0.481i)16-s + (0.844 + 0.535i)17-s + (−0.0627 − 0.998i)19-s + (−0.587 + 0.809i)22-s + (−0.904 + 0.425i)23-s + (0.809 + 0.587i)26-s + (0.982 − 0.187i)28-s + (0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (−0.125 − 0.992i)2-s + (−0.968 + 0.248i)4-s + (−0.998 − 0.0627i)7-s + (0.368 + 0.929i)8-s + (−0.728 − 0.684i)11-s + (−0.684 + 0.728i)13-s + (0.0627 + 0.998i)14-s + (0.876 − 0.481i)16-s + (0.844 + 0.535i)17-s + (−0.0627 − 0.998i)19-s + (−0.587 + 0.809i)22-s + (−0.904 + 0.425i)23-s + (0.809 + 0.587i)26-s + (0.982 − 0.187i)28-s + (0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.283 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6015\)    =    \(3 \cdot 5 \cdot 401\)
Sign: $0.283 + 0.958i$
Analytic conductor: \(27.9335\)
Root analytic conductor: \(27.9335\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6015} (3833, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6015,\ (0:\ ),\ 0.283 + 0.958i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07390318854 + 0.05519795577i\)
\(L(\frac12)\) \(\approx\) \(0.07390318854 + 0.05519795577i\)
\(L(1)\) \(\approx\) \(0.5666034416 - 0.3126775528i\)
\(L(1)\) \(\approx\) \(0.5666034416 - 0.3126775528i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
401 \( 1 \)
good2 \( 1 + (-0.125 - 0.992i)T \)
7 \( 1 + (-0.998 - 0.0627i)T \)
11 \( 1 + (-0.728 - 0.684i)T \)
13 \( 1 + (-0.684 + 0.728i)T \)
17 \( 1 + (0.844 + 0.535i)T \)
19 \( 1 + (-0.0627 - 0.998i)T \)
23 \( 1 + (-0.904 + 0.425i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (-0.637 + 0.770i)T \)
37 \( 1 + (-0.844 - 0.535i)T \)
41 \( 1 + (0.929 - 0.368i)T \)
43 \( 1 + (-0.982 - 0.187i)T \)
47 \( 1 + (0.770 - 0.637i)T \)
53 \( 1 + (0.998 - 0.0627i)T \)
59 \( 1 + (-0.425 - 0.904i)T \)
61 \( 1 + (0.535 - 0.844i)T \)
67 \( 1 + (0.248 - 0.968i)T \)
71 \( 1 + (-0.728 - 0.684i)T \)
73 \( 1 + (0.684 + 0.728i)T \)
79 \( 1 + (-0.876 + 0.481i)T \)
83 \( 1 + (0.951 - 0.309i)T \)
89 \( 1 + (0.968 + 0.248i)T \)
97 \( 1 + (0.844 - 0.535i)T \)
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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

−17.504937713794586269291388120158, −16.764330230601306597552407190440, −16.199164159119403054328416623765, −15.81602876989059167441748476455, −14.86932215522869314930696267137, −14.62047318202221766010349681905, −13.68150114858670256296303411678, −13.055789012363287180416961580274, −12.42932115132199094274516248857, −12.01058104601290227981660900537, −10.45456684968008001864996114197, −10.18018803307365921824084888363, −9.63631135067066524973436432130, −8.842030283499602180609760684705, −7.999559034305601320500142024500, −7.47012771398989153689261516096, −6.93385818278400122297837858524, −5.962328596813900580311185029611, −5.56899907899458283207241813253, −4.80550405569717445271758763171, −3.95887482661930767068823604857, −3.18122249308703561665583974410, −2.365911352674976027122011916287, −1.132165454870990243815244624436, −0.03514650622976651324713490211, 0.75968947773111645133304887538, 1.991862434989250465242341611, 2.50182098637455981870492965569, 3.45324080368258740633227415035, 3.78619773043924220904418175311, 4.84391884222371990069802219888, 5.486367236714495448749700712195, 6.27882537446957964506421165569, 7.22323500230888440647436424996, 7.898219405476154254890017391679, 8.75257821898244054029137648546, 9.29051519027528322666309359270, 10.04850248822993568651730454025, 10.43350193867472907210144621372, 11.243714543289319527276951735090, 11.975535154270197959497931801992, 12.49718659208460294292870773107, 13.14002210007846501347020155352, 13.81742476077508630492245633047, 14.22750850224622072484617814189, 15.282221004709874227654080874931, 15.9806628822001599731864379091, 16.65706007143260641654730340065, 17.2508853777632608578553276535, 17.96427623520930479685946483895

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