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Properties

Label 1-2013-2013.107-r0-0-0
Degree $1$
Conductor $2013$
Sign $-0.562 - 0.826i$
Analytic cond. $9.34833$
Root an. cond. $9.34833$
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  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s + (0.978 − 0.207i)5-s + (−0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s + (−0.104 − 0.994i)10-s + (0.978 − 0.207i)13-s + (0.104 + 0.994i)14-s + (0.913 + 0.406i)16-s + (−0.669 − 0.743i)17-s + (0.978 + 0.207i)19-s − 20-s + (−0.809 − 0.587i)23-s + (0.913 − 0.406i)25-s + (−0.104 − 0.994i)26-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)2-s + (−0.978 − 0.207i)4-s + (0.978 − 0.207i)5-s + (−0.978 + 0.207i)7-s + (−0.309 + 0.951i)8-s + (−0.104 − 0.994i)10-s + (0.978 − 0.207i)13-s + (0.104 + 0.994i)14-s + (0.913 + 0.406i)16-s + (−0.669 − 0.743i)17-s + (0.978 + 0.207i)19-s − 20-s + (−0.809 − 0.587i)23-s + (0.913 − 0.406i)25-s + (−0.104 − 0.994i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.562 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2013\)    =    \(3 \cdot 11 \cdot 61\)
Sign: $-0.562 - 0.826i$
Analytic conductor: \(9.34833\)
Root analytic conductor: \(9.34833\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2013} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2013,\ (0:\ ),\ -0.562 - 0.826i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7164122092 - 1.353695803i\)
\(L(\frac12)\) \(\approx\) \(0.7164122092 - 1.353695803i\)
\(L(1)\) \(\approx\) \(0.9050098213 - 0.6233120282i\)
\(L(1)\) \(\approx\) \(0.9050098213 - 0.6233120282i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.104 - 0.994i)T \)
5 \( 1 + (0.978 - 0.207i)T \)
7 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (0.978 - 0.207i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (0.978 + 0.207i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.913 + 0.406i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.913 + 0.406i)T \)
47 \( 1 + (0.104 + 0.994i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (0.104 - 0.994i)T \)
71 \( 1 + (0.913 - 0.406i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−20.149745671408238437356194971766, −19.27914444770994937387899859687, −18.49699173883288631786272871771, −17.86371533756575893351618921071, −17.28177412083100955262605703989, −16.449333599661814164265963490786, −15.88131532795937533659416767365, −15.22738573754183032421917500182, −14.201050231152027366259744072160, −13.6457102638437252114287269743, −13.18820165800170318341923805342, −12.43572246251905247559789690403, −11.23886296793156160027750944932, −10.221954243744986586023125190703, −9.6681711423526726402072163691, −8.96376788377904838472405275167, −8.211518515384876375143235368472, −7.0347121972781533657520597559, −6.62910777808976171663812745558, −5.798513095240731317170784341394, −5.316607773247446284194583722915, −3.95534200243320788821393562569, −3.48511940248671326310310960632, −2.22785750217137566377199861734, −1.00937677887049248993632366385, 0.60241309348852124700065671588, 1.63763881224570450224315643123, 2.53960453011002620490332948711, 3.22947287206196152023025148018, 4.144932178998476919417023857069, 5.15116158630428817591692153010, 5.89682725646008645959105958078, 6.51607937814805998382005273159, 7.82961797921252923732269906135, 8.92798110085363947957466269713, 9.29577272102972892088950589660, 10.05501725148343540573997493416, 10.676543009675021448613597189718, 11.61056925586398113080698990841, 12.35757391654109535583295441668, 13.12592992480146728260949412770, 13.61786824879339925859925160816, 14.140093306178925148814195058819, 15.300688934555608866604512267965, 16.11762847262750056555984575894, 16.88391239201466159739406960367, 17.77826396431422004865077214575, 18.42415512342784646825302660097, 18.80714517520051211036221996904, 19.924421720292133020431908544973

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