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Properties

Label 1-2015-2015.1127-r0-0-0
Degree $1$
Conductor $2015$
Sign $-0.840 + 0.541i$
Analytic cond. $9.35762$
Root an. cond. $9.35762$
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  + (−0.406 + 0.913i)2-s + (0.406 + 0.913i)3-s + (−0.669 − 0.743i)4-s − 6-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.669 + 0.743i)9-s + (−0.309 + 0.951i)11-s + (0.406 − 0.913i)12-s + (−0.669 + 0.743i)14-s + (−0.104 + 0.994i)16-s + (0.951 − 0.309i)17-s + (−0.406 − 0.913i)18-s + (0.809 − 0.587i)19-s + (0.104 + 0.994i)21-s + (−0.743 − 0.669i)22-s + ⋯
L(s)  = 1  + (−0.406 + 0.913i)2-s + (0.406 + 0.913i)3-s + (−0.669 − 0.743i)4-s − 6-s + (0.951 + 0.309i)7-s + (0.951 − 0.309i)8-s + (−0.669 + 0.743i)9-s + (−0.309 + 0.951i)11-s + (0.406 − 0.913i)12-s + (−0.669 + 0.743i)14-s + (−0.104 + 0.994i)16-s + (0.951 − 0.309i)17-s + (−0.406 − 0.913i)18-s + (0.809 − 0.587i)19-s + (0.104 + 0.994i)21-s + (−0.743 − 0.669i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.840 + 0.541i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ -0.840 + 0.541i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4857237714 + 1.650627932i\)
\(L(\frac12)\) \(\approx\) \(0.4857237714 + 1.650627932i\)
\(L(1)\) \(\approx\) \(0.7742199904 + 0.8076693823i\)
\(L(1)\) \(\approx\) \(0.7742199904 + 0.8076693823i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
13 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.406 + 0.913i)T \)
3 \( 1 + (0.406 + 0.913i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (0.951 - 0.309i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 + (0.743 + 0.669i)T \)
29 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.866 + 0.5i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (0.587 - 0.809i)T \)
53 \( 1 + (0.207 - 0.978i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (-0.743 + 0.669i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (0.994 - 0.104i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (-0.743 + 0.669i)T \)
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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

−19.55154177365397240140667234276, −18.87156180389899988413003768242, −18.39694415213095202197362813789, −17.73759668626414213404314633482, −16.9697416960790054061500395995, −16.340946154440709742331457227799, −14.97222213167665070954971187120, −14.108702687808269804307850237100, −13.815103495655059162976638286608, −12.9029628768174335857807943733, −12.19505322397769346602456366020, −11.543462024879398430692547877843, −10.86122900658864967285407804478, −10.05035661404394161709654359091, −9.07250753031027497542205046446, −8.24725716371385546927717102012, −7.94496312344617838177294802344, −7.121625618921201946508829157686, −5.9202351081726680692102222339, −5.04087651773073386699037549768, −3.92406703747091815369730636437, −3.111254083842815308969307987385, −2.38859499509075688866389192699, −1.29759034564499605220935625900, −0.83073094200689853905831773035, 1.120767084370732969688235088347, 2.2311701016319713150105025627, 3.339411762049801309851974524165, 4.444431027354665150630501539732, 5.15618533760414103851797416558, 5.41419490374535671474115380611, 6.84354407992114171796304203222, 7.58617436046120376120243682099, 8.25794602441230758990482196017, 8.95184843313709683647942321735, 9.77414145435087457336273872436, 10.21496613854186071500475441310, 11.20214072504824950350073923599, 11.9622990325889402139880676008, 13.26230611769148956906375863901, 13.9298454010028739729993106245, 14.73358336608795063805145586498, 15.10926379159244251616294713034, 15.76663584698572604050427195621, 16.52921343456612264313628864741, 17.2624102096357681182315155334, 17.92366931143181151073361348014, 18.57755210877078491938030160839, 19.506909912725488614493609145752, 20.27146624858604142649555403187

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