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Properties

Label 1-1015-1015.713-r1-0-0
Degree $1$
Conductor $1015$
Sign $0.677 + 0.735i$
Analytic cond. $109.076$
Root an. cond. $109.076$
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  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − i·11-s − 12-s + i·13-s + 16-s + 17-s − 18-s − i·19-s + i·22-s + i·23-s + 24-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − i·11-s − 12-s + i·13-s + 16-s + 17-s − 18-s − i·19-s + i·22-s + i·23-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.677 + 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1015\)    =    \(5 \cdot 7 \cdot 29\)
Sign: $0.677 + 0.735i$
Analytic conductor: \(109.076\)
Root analytic conductor: \(109.076\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1015} (713, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1015,\ (1:\ ),\ 0.677 + 0.735i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7493350078 + 0.3287932974i\)
\(L(\frac12)\) \(\approx\) \(0.7493350078 + 0.3287932974i\)
\(L(1)\) \(\approx\) \(0.5573352051 + 0.02318212310i\)
\(L(1)\) \(\approx\) \(0.5573352051 + 0.02318212310i\)

Euler product

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

−21.00337471022137372516414888549, −20.6368822823959181077261975909, −19.63671830543345802709862898097, −18.67329908584404472195020956268, −18.150143514507212956335879970117, −17.485175684153324108438650900051, −16.68047279476549244545212888947, −16.20549065977253974446459608707, −15.16574371419143981080962146616, −14.61530987262608488163329078735, −12.89802220818213908461755391831, −12.42627841596238823164444555759, −11.61940272091367945205571625206, −10.745735591136614334733186160293, −10.00462020969577912102379361544, −9.5897363539306939865945292348, −8.0717039599298116701437536627, −7.65085142962670901426522110927, −6.573946178753332378680704727253, −5.88696557385978875559285194653, −4.96063454262787522963351364166, −3.7231416720266244861597556579, −2.424416032236471929982991752668, −1.33585886989903971450886436335, −0.42751545191505677758627057918, 0.73648797110986307141469532412, 1.50655196269109328908265339647, 2.8369083981482758375602773034, 3.98959529050937983758545601828, 5.31400780694813728220526425142, 6.01260694379904104718904871390, 6.906710546512150175909410926741, 7.572996968510411677564927780, 8.71448027813713337679370693846, 9.46737774545999233546482425022, 10.31764200521597815102041474500, 11.18311527955109147026499451075, 11.60693500743725188179698588421, 12.44280377861088315884551435661, 13.52865761463977364907938407066, 14.5777093648179865641128312364, 15.72346633795176141368101032043, 16.21446786274625768246210059484, 16.8898021393484896555634004476, 17.58601369484593718164547058860, 18.33643728081669234233509845180, 19.1156614138191513148788561440, 19.56470178692497863446357906579, 20.91592758986256956051150825845, 21.454648583279436187059514732722

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