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Properties

Label 1-1013-1013.45-r1-0-0
Degree $1$
Conductor $1013$
Sign $-0.372 + 0.928i$
Analytic cond. $108.861$
Root an. cond. $108.861$
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  + i·2-s − i·3-s − 4-s − i·5-s + 6-s − i·7-s − i·8-s − 9-s + 10-s − 11-s + i·12-s − 13-s + 14-s − 15-s + 16-s − i·17-s + ⋯
L(s)  = 1  + i·2-s − i·3-s − 4-s − i·5-s + 6-s − i·7-s − i·8-s − 9-s + 10-s − 11-s + i·12-s − 13-s + 14-s − 15-s + 16-s − i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1013 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1013 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1013\)
Sign: $-0.372 + 0.928i$
Analytic conductor: \(108.861\)
Root analytic conductor: \(108.861\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1013} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1013,\ (1:\ ),\ -0.372 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3167617737 - 0.4684139638i\)
\(L(\frac12)\) \(\approx\) \(-0.3167617737 - 0.4684139638i\)
\(L(1)\) \(\approx\) \(0.6418234002 - 0.2742444674i\)
\(L(1)\) \(\approx\) \(0.6418234002 - 0.2742444674i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1013 \( 1 \)
good2 \( 1 \)
3 \( 1 + T \)
5 \( 1 + iT \)
7 \( 1 - iT \)
11 \( 1 - T \)
13 \( 1 - iT \)
17 \( 1 + T \)
19 \( 1 - iT \)
23 \( 1 - iT \)
29 \( 1 - T \)
31 \( 1 + T \)
37 \( 1 - T \)
41 \( 1 + iT \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 - iT \)
67 \( 1 - iT \)
71 \( 1 + T \)
73 \( 1 + iT \)
79 \( 1 - T \)
83 \( 1 - iT \)
89 \( 1 - T \)
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

−21.73038422424848832326591745081, −21.548601684392177870023471110224, −20.46253178767999701560474325592, −19.66358775318708461641556096637, −19.01968801832545274831376142138, −17.931702820267497854286973004914, −17.79811909626960581446680674095, −16.3568433416663638552717750431, −15.56891344548353549512743681547, −14.627394213343216595100257454360, −14.3671499355635926461476836816, −13.116624761440457869445736446781, −12.1412687359818171635238532542, −11.52888411191200191073780513738, −10.56622232429154092642752309334, −10.160957580855436805365903224108, −9.37270227879199138409450716837, −8.457886807340262472586164880648, −7.57739913994507764556338089173, −5.97584574940795130328902777656, −5.32059159453269498932001952444, −4.4333595484162157010386792251, −3.22045084629321045029560121089, −2.84266728735015196446817108287, −1.86809081812340334748445082587, 0.20395277096989853895183627700, 0.55550527540525527285848700697, 1.91839117239393032061784371390, 3.32352752283224781101198756531, 4.63525237141386222104381754090, 5.2002557403493387445932495698, 6.13200512015272238131442676114, 7.22889765357641779376899970138, 7.71876748740764844673564656110, 8.26824048689304697450781568094, 9.48970248909942719442213598476, 10.06079730345565285686395327945, 11.61311997614761389609799631210, 12.3302607568908354123230774346, 13.360641337872018166921370907845, 13.543927168763260205418327833989, 14.37676910091854627313437405197, 15.55348470187681742217712921479, 16.29684308603271843684157689736, 17.04762939618379349305154538190, 17.54589556255918472007090416588, 18.38062442061859792538638222864, 19.14315597223840532144332838548, 20.2191453416265197447153752608, 20.53691416721655920862495049018

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