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Properties

Label 1-6015-6015.1208-r0-0-0
Degree $1$
Conductor $6015$
Sign $0.916 - 0.399i$
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  + (−0.684 + 0.728i)2-s + (−0.0627 − 0.998i)4-s + (0.368 + 0.929i)7-s + (0.770 + 0.637i)8-s + (0.187 − 0.982i)11-s + (−0.982 − 0.187i)13-s + (−0.929 − 0.368i)14-s + (−0.992 + 0.125i)16-s + (0.248 + 0.968i)17-s + (0.929 + 0.368i)19-s + (0.587 + 0.809i)22-s + (−0.481 − 0.876i)23-s + (0.809 − 0.587i)26-s + (0.904 − 0.425i)28-s + (0.309 + 0.951i)29-s + ⋯
L(s)  = 1  + (−0.684 + 0.728i)2-s + (−0.0627 − 0.998i)4-s + (0.368 + 0.929i)7-s + (0.770 + 0.637i)8-s + (0.187 − 0.982i)11-s + (−0.982 − 0.187i)13-s + (−0.929 − 0.368i)14-s + (−0.992 + 0.125i)16-s + (0.248 + 0.968i)17-s + (0.929 + 0.368i)19-s + (0.587 + 0.809i)22-s + (−0.481 − 0.876i)23-s + (0.809 − 0.587i)26-s + (0.904 − 0.425i)28-s + (0.309 + 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.916 - 0.399i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6015\)    =    \(3 \cdot 5 \cdot 401\)
Sign: $0.916 - 0.399i$
Analytic conductor: \(27.9335\)
Root analytic conductor: \(27.9335\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6015} (1208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6015,\ (0:\ ),\ 0.916 - 0.399i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9403530230 - 0.1962238829i\)
\(L(\frac12)\) \(\approx\) \(0.9403530230 - 0.1962238829i\)
\(L(1)\) \(\approx\) \(0.7332306667 + 0.1802251327i\)
\(L(1)\) \(\approx\) \(0.7332306667 + 0.1802251327i\)

Euler product

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

−17.74706242183032326436630524127, −17.38406920649280298373175828413, −16.56921527412063802624525957160, −16.06560074007149803361805724436, −15.16822099861164390696768760715, −14.356870049974851194324928397548, −13.66496887555526567991001734008, −13.18221793471974117907310992699, −12.1819066483606669723831979420, −11.73530481495041277956592482826, −11.287202635130047609142975766803, −10.23002547969741210611781129656, −9.6915953767936985397351562969, −9.59109537781375946704815713598, −8.27804656876646011201020074419, −7.83468620491715255286309352182, −7.012198370350843424685306784431, −6.8132221991882062458721055328, −5.1793112806815731045673106022, −4.71857453628509202557944923208, −3.97282471823244928915656731041, −3.12661074473304769861712134465, −2.406474275164199772952275807942, −1.5308497027192347936035089718, −0.86390947625883318091003259637, 0.37621101381097962870444098388, 1.432066740921916628610220755767, 2.194202195344115137176211974709, 3.03992400103184976418444390316, 4.09435467465353638283141662401, 5.06459875450130422070993764918, 5.56130991736342247609228021392, 6.17228098023992383400769740189, 6.88114319369980508794146423852, 7.94232727237445608579569182695, 8.09993947965926241953156611499, 8.981106781855560793389484602030, 9.46658192397358127679425645497, 10.33244368444655040016803724695, 10.8467237988189162722395564694, 11.73884434533220122110794389156, 12.27276853028388639253517571827, 13.13194051868886105737682427928, 14.17981868307831001844503317539, 14.43562857241892670979542617175, 15.07959552849212370569728645131, 15.86992591518252650569352605317, 16.307744620030434075790222200017, 17.05842986098815213969785973180, 17.60572848137124164697137359764

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