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Properties

Label 1-2015-2015.1139-r0-0-0
Degree $1$
Conductor $2015$
Sign $-0.940 + 0.339i$
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  + (−0.587 + 0.809i)2-s + (−0.809 + 0.587i)3-s + (−0.309 − 0.951i)4-s − i·6-s + (−0.951 + 0.309i)7-s + (0.951 + 0.309i)8-s + (0.309 − 0.951i)9-s + (−0.951 + 0.309i)11-s + (0.809 + 0.587i)12-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s + (0.587 + 0.809i)18-s + (0.587 − 0.809i)19-s + (0.587 − 0.809i)21-s + (0.309 − 0.951i)22-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (−0.809 + 0.587i)3-s + (−0.309 − 0.951i)4-s − i·6-s + (−0.951 + 0.309i)7-s + (0.951 + 0.309i)8-s + (0.309 − 0.951i)9-s + (−0.951 + 0.309i)11-s + (0.809 + 0.587i)12-s + (0.309 − 0.951i)14-s + (−0.809 + 0.587i)16-s + (−0.309 + 0.951i)17-s + (0.587 + 0.809i)18-s + (0.587 − 0.809i)19-s + (0.587 − 0.809i)21-s + (0.309 − 0.951i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.940 + 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.940 + 0.339i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ -0.940 + 0.339i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07732320978 + 0.4417633733i\)
\(L(\frac12)\) \(\approx\) \(0.07732320978 + 0.4417633733i\)
\(L(1)\) \(\approx\) \(0.4107342477 + 0.2763043910i\)
\(L(1)\) \(\approx\) \(0.4107342477 + 0.2763043910i\)

Euler product

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

−19.4723984173000656390802966314, −18.70008961919702942986960635314, −18.395437025387006264470818367483, −17.61239223764593737393896706095, −16.79299866455746256073001170083, −16.18287521505974701248760757931, −15.74935414445078658168763827871, −14.00595341581383832218433409412, −13.524010918025535260838563123359, −12.74708247731510331558422739097, −12.23615517559025643437014259654, −11.46105704039232711293219318888, −10.67220792763656584863193258885, −10.110432827314132517683025229524, −9.4185052625560926891497028670, −8.24301423187496947258425208147, −7.6766775502378965200556921092, −6.83055546361832217073878243457, −6.06338249810560400057108994643, −5.00817330730704281572249260155, −4.170188216361054464617178076248, −2.96746881982760326888642053856, −2.44237279022581611245757860298, −1.15617262607794445543498954424, −0.336020491247275683677842868795, 0.775448856115052436482433255380, 2.155377656367026026113163075982, 3.41010755317785728638621141417, 4.40313055468260913074997756433, 5.32877567856981773312600386434, 5.78270254015051461455103063784, 6.66180309778141888848377859901, 7.27444011440497275732506697345, 8.318038272531710220605795507913, 9.2504423414784798927913123669, 9.714614192674256744150725327108, 10.52744012138338380263952370566, 11.03140514863892574616020520430, 12.1695492457944515327738015391, 12.92651395750106316491406125043, 13.70013933616087141025338164837, 14.82717127454130051949326852161, 15.501340905361996545122328360753, 15.91841077613233597296967613059, 16.459428587438912026080715340337, 17.42772024367978814470308942881, 17.87486315276896352047207707480, 18.50383108254078591830765933012, 19.557642931162494670804994898675, 19.90991229841638075334418223317

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