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Properties

Label 1-2015-2015.1039-r0-0-0
Degree $1$
Conductor $2015$
Sign $-0.800 - 0.599i$
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  + (0.309 − 0.951i)2-s + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s − 6-s + (−0.809 − 0.587i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (−0.309 + 0.951i)12-s + (−0.809 + 0.587i)14-s + (0.309 + 0.951i)16-s + (0.809 − 0.587i)17-s + (0.309 + 0.951i)18-s + (−0.309 + 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.809 − 0.587i)22-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)4-s − 6-s + (−0.809 − 0.587i)7-s + (−0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (0.809 + 0.587i)11-s + (−0.309 + 0.951i)12-s + (−0.809 + 0.587i)14-s + (0.309 + 0.951i)16-s + (0.809 − 0.587i)17-s + (0.309 + 0.951i)18-s + (−0.309 + 0.951i)19-s + (−0.309 + 0.951i)21-s + (0.809 − 0.587i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.800 - 0.599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2015\)    =    \(5 \cdot 13 \cdot 31\)
Sign: $-0.800 - 0.599i$
Analytic conductor: \(9.35762\)
Root analytic conductor: \(9.35762\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2015} (1039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2015,\ (0:\ ),\ -0.800 - 0.599i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4280327699 - 1.284877092i\)
\(L(\frac12)\) \(\approx\) \(0.4280327699 - 1.284877092i\)
\(L(1)\) \(\approx\) \(0.6641781819 - 0.7354575060i\)
\(L(1)\) \(\approx\) \(0.6641781819 - 0.7354575060i\)

Euler product

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

−20.29435843889492506558638521860, −19.399641670700368109176922122, −18.71521764427376476323881984536, −17.75375549414299358807413828900, −16.89906238293524224129091232372, −16.66618577518280597301754858868, −15.77025266485141915378740949980, −15.24885152023808178443605497648, −14.63606155784439355576183479929, −13.82254380115872889509104748990, −12.9629950973081534615544555208, −12.169108106366688778453152903790, −11.48238278989670483053604332358, −10.42862616557925146151228459499, −9.60407369372776985584841296520, −8.877094585754429231725479220060, −8.54814340991304844533830669630, −7.13360816101761623866538486741, −6.50341749453606261600245723392, −5.6481522039349181444781015199, −5.24557812341972081579497599203, −4.083151797033287477309269931975, −3.52271850784765803982355833995, −2.740741691458785563257836780, −0.78021797266935731513646578733, 0.691603662591855156451049154486, 1.39061391880306397908573075524, 2.43919618919687776564750896085, 3.225250535526165596097065349827, 4.15039992820649817603170684337, 5.00889783913885335887844969959, 6.092519852024849609233422724624, 6.56000162008595372194045063630, 7.57045439837226190887998706008, 8.40387953437338733897310016428, 9.53688232816515401313628479556, 9.95618679198722359478437196883, 10.94901093759400176277018246821, 11.65685573827074454152480459471, 12.35654778760717917969225765072, 12.86959099200809844846961737180, 13.54171820299408483359391819770, 14.33830139586228325644589953748, 14.83382765625287979387191049722, 16.25542191619025974363493277069, 16.935466663963633296258269940215, 17.57352146818224953139720859642, 18.49642760156381199515352317215, 18.99369713410762820628185517119, 19.611610129625093050873775169508

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