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Properties

Label 1-2557-2557.2-r1-0-0
Degree $1$
Conductor $2557$
Sign $-0.311 - 0.950i$
Analytic cond. $274.787$
Root an. cond. $274.787$
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  + (0.999 + 0.00245i)2-s + (0.907 − 0.419i)3-s + (0.999 + 0.00491i)4-s + (0.898 − 0.439i)5-s + (0.908 − 0.417i)6-s + (0.302 − 0.953i)7-s + (0.999 + 0.00737i)8-s + (0.648 − 0.761i)9-s + (0.899 − 0.437i)10-s + (0.753 − 0.657i)11-s + (0.909 − 0.414i)12-s + (0.0859 − 0.996i)13-s + (0.304 − 0.952i)14-s + (0.631 − 0.775i)15-s + (0.999 + 0.00983i)16-s + (−0.997 − 0.0761i)17-s + ⋯
L(s)  = 1  + (0.999 + 0.00245i)2-s + (0.907 − 0.419i)3-s + (0.999 + 0.00491i)4-s + (0.898 − 0.439i)5-s + (0.908 − 0.417i)6-s + (0.302 − 0.953i)7-s + (0.999 + 0.00737i)8-s + (0.648 − 0.761i)9-s + (0.899 − 0.437i)10-s + (0.753 − 0.657i)11-s + (0.909 − 0.414i)12-s + (0.0859 − 0.996i)13-s + (0.304 − 0.952i)14-s + (0.631 − 0.775i)15-s + (0.999 + 0.00983i)16-s + (−0.997 − 0.0761i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2557\)
Sign: $-0.311 - 0.950i$
Analytic conductor: \(274.787\)
Root analytic conductor: \(274.787\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2557} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2557,\ (1:\ ),\ -0.311 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.793426645 - 7.992509723i\)
\(L(\frac12)\) \(\approx\) \(5.793426645 - 7.992509723i\)
\(L(1)\) \(\approx\) \(3.114591783 - 1.578789899i\)
\(L(1)\) \(\approx\) \(3.114591783 - 1.578789899i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2557 \( 1 \)
good2 \( 1 + (0.999 + 0.00245i)T \)
3 \( 1 + (0.907 - 0.419i)T \)
5 \( 1 + (0.898 - 0.439i)T \)
7 \( 1 + (0.302 - 0.953i)T \)
11 \( 1 + (0.753 - 0.657i)T \)
13 \( 1 + (0.0859 - 0.996i)T \)
17 \( 1 + (-0.997 - 0.0761i)T \)
19 \( 1 + (-0.188 + 0.982i)T \)
23 \( 1 + (-0.991 + 0.127i)T \)
29 \( 1 + (-0.967 - 0.252i)T \)
31 \( 1 + (-0.0344 + 0.999i)T \)
37 \( 1 + (0.844 + 0.535i)T \)
41 \( 1 + (0.798 + 0.602i)T \)
43 \( 1 + (-0.728 - 0.685i)T \)
47 \( 1 + (0.996 - 0.0859i)T \)
53 \( 1 + (-0.487 + 0.873i)T \)
59 \( 1 + (0.358 - 0.933i)T \)
61 \( 1 + (0.325 + 0.945i)T \)
67 \( 1 + (0.789 + 0.614i)T \)
71 \( 1 + (-0.652 - 0.758i)T \)
73 \( 1 + (0.149 - 0.988i)T \)
79 \( 1 + (0.970 + 0.240i)T \)
83 \( 1 + (-0.0245 - 0.999i)T \)
89 \( 1 + (-0.776 + 0.629i)T \)
97 \( 1 + (-0.972 + 0.233i)T \)
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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

−19.66401624689558948313387099896, −18.89412889023992126522837159911, −18.094535515894473352681951400597, −17.22206626984320717489943786525, −16.36839301182049071571055951604, −15.53402144197554272585901155198, −14.981913628847778177394448217847, −14.44508667415646003051621258549, −13.9053684634455987772556419005, −13.13950916531897541245443102309, −12.53675631650898941828033998423, −11.37534977053172002031819092337, −11.0821170130193971745735647032, −9.85229908518725883123144116766, −9.354640090440192240492657047389, −8.65958131234328736569097908949, −7.52909987942626361451700613707, −6.75780133809762804262188479442, −6.11792275654918618403787417914, −5.19386272392489544179659208395, −4.36967885273815166056027065748, −3.82582848738383699162454164136, −2.545340830983262207317514948407, −2.2326173968105508963859595134, −1.63327335141967180092434258201, 0.80664884509121753563767048243, 1.54219959624377975544668596101, 2.247608193811463792879432877999, 3.27097463928855017754920849898, 3.93463628315500452626511363791, 4.65525883523157606043849265713, 5.797391872171260222968011484699, 6.29860694933266970482456243325, 7.16365126389161017451014592377, 7.9738380740388641838825004721, 8.60880188449391472541299232651, 9.66778387201679462679826067093, 10.34737826701048609072043290324, 11.131515608306296944736515102540, 12.14472745765210677646629956422, 12.83465926542254483208194903724, 13.478357809417089131544618889181, 13.87893124774552975193320317443, 14.47036368011788979567060145359, 15.15741343248621231004112399214, 16.12654679007475802028157328606, 16.78615238833179802928622940056, 17.551638757516522326885353141964, 18.26805325907227410087778852971, 19.36781872453472038626630377096

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