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Properties

Label 1-1020-1020.479-r1-0-0
Degree $1$
Conductor $1020$
Sign $0.982 + 0.185i$
Analytic cond. $109.614$
Root an. cond. $109.614$
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  + (−0.382 − 0.923i)7-s + (−0.923 + 0.382i)11-s − i·13-s + (−0.707 + 0.707i)19-s + (−0.923 + 0.382i)23-s + (−0.382 + 0.923i)29-s + (0.923 + 0.382i)31-s + (−0.923 − 0.382i)37-s + (0.382 + 0.923i)41-s + (0.707 + 0.707i)43-s − i·47-s + (−0.707 + 0.707i)49-s + (0.707 − 0.707i)53-s + (−0.707 − 0.707i)59-s + (0.382 + 0.923i)61-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)7-s + (−0.923 + 0.382i)11-s − i·13-s + (−0.707 + 0.707i)19-s + (−0.923 + 0.382i)23-s + (−0.382 + 0.923i)29-s + (0.923 + 0.382i)31-s + (−0.923 − 0.382i)37-s + (0.382 + 0.923i)41-s + (0.707 + 0.707i)43-s − i·47-s + (−0.707 + 0.707i)49-s + (0.707 − 0.707i)53-s + (−0.707 − 0.707i)59-s + (0.382 + 0.923i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 17\)
Sign: $0.982 + 0.185i$
Analytic conductor: \(109.614\)
Root analytic conductor: \(109.614\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1020} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1020,\ (1:\ ),\ 0.982 + 0.185i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.261801290 + 0.1178380059i\)
\(L(\frac12)\) \(\approx\) \(1.261801290 + 0.1178380059i\)
\(L(1)\) \(\approx\) \(0.8812937645 - 0.06020544315i\)
\(L(1)\) \(\approx\) \(0.8812937645 - 0.06020544315i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
17 \( 1 \)
good7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (-0.923 + 0.382i)T \)
13 \( 1 - iT \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.923 + 0.382i)T \)
29 \( 1 + (-0.382 + 0.923i)T \)
31 \( 1 + (0.923 + 0.382i)T \)
37 \( 1 + (-0.923 - 0.382i)T \)
41 \( 1 + (0.382 + 0.923i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.707 - 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (0.382 + 0.923i)T \)
67 \( 1 - T \)
71 \( 1 + (0.923 + 0.382i)T \)
73 \( 1 + (0.382 - 0.923i)T \)
79 \( 1 + (0.923 - 0.382i)T \)
83 \( 1 + (0.707 - 0.707i)T \)
89 \( 1 + iT \)
97 \( 1 + (-0.382 + 0.923i)T \)
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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

−21.26371243668747330418112230021, −20.90238142206916909617738239937, −19.64337196132985280233138695446, −18.95274299704734173576203507515, −18.48658657863655713497440865767, −17.47673198073579680850717897831, −16.65710296427480732283525948225, −15.65237204091687874893151805747, −15.42135188752193374081981837612, −14.16690019683212757955136450995, −13.51289748281072180350477015299, −12.5692195696846510285874593395, −11.90069680828352453826505114478, −11.01253715905050786963406336065, −10.12328847462861049879229682097, −9.18923740682321918874419874689, −8.5367181257689062265054211195, −7.605841564303026032509111604753, −6.48001891799900733349303493864, −5.85799059986409807116075490621, −4.83923321494305725665939461871, −3.88925077690203618410323574563, −2.63440009236098698819392149989, −2.088342817245776369984858822470, −0.42663806380857086314856944422, 0.590986746761029949402164322988, 1.84651027958804849350781506375, 3.01947901125901786453464030057, 3.8560280332450633071218093425, 4.871750410355973169200545333313, 5.791245732193752279203439728677, 6.75863753052451553434372320832, 7.6940226656900499169641216639, 8.23708726777088539399831052350, 9.51334032005892789219975406742, 10.413817565424416616416931212948, 10.64645883152243337657832490207, 12.00502163554489177188706943588, 12.81442463606320451570099719629, 13.39233209406818673670459280655, 14.310375982647656515427067593024, 15.17521235011687508701885713597, 16.00014253180855089686494075863, 16.65915090696826031248624724715, 17.66676386529022613446093483892, 18.118594597290172875171607497951, 19.2477393958907856720987368523, 19.89589021876794520084663690216, 20.639413528358413370633508218, 21.27750545233400425975443055458

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