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Properties

Label 1-2557-2557.1082-r0-0-0
Degree $1$
Conductor $2557$
Sign $-0.995 + 0.0946i$
Analytic cond. $11.8746$
Root an. cond. $11.8746$
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  + (0.516 + 0.856i)2-s + (0.202 + 0.979i)3-s + (−0.465 + 0.885i)4-s + (−0.0270 − 0.999i)5-s + (−0.733 + 0.679i)6-s + (0.934 − 0.355i)7-s + (−0.998 + 0.0589i)8-s + (−0.917 + 0.396i)9-s + (0.841 − 0.539i)10-s + (0.0172 + 0.999i)11-s + (−0.960 − 0.276i)12-s + (0.163 + 0.986i)13-s + (0.787 + 0.616i)14-s + (0.973 − 0.229i)15-s + (−0.566 − 0.824i)16-s + (0.905 + 0.423i)17-s + ⋯
L(s)  = 1  + (0.516 + 0.856i)2-s + (0.202 + 0.979i)3-s + (−0.465 + 0.885i)4-s + (−0.0270 − 0.999i)5-s + (−0.733 + 0.679i)6-s + (0.934 − 0.355i)7-s + (−0.998 + 0.0589i)8-s + (−0.917 + 0.396i)9-s + (0.841 − 0.539i)10-s + (0.0172 + 0.999i)11-s + (−0.960 − 0.276i)12-s + (0.163 + 0.986i)13-s + (0.787 + 0.616i)14-s + (0.973 − 0.229i)15-s + (−0.566 − 0.824i)16-s + (0.905 + 0.423i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2557\)
Sign: $-0.995 + 0.0946i$
Analytic conductor: \(11.8746\)
Root analytic conductor: \(11.8746\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2557} (1082, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2557,\ (0:\ ),\ -0.995 + 0.0946i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1066920736 + 2.249234813i\)
\(L(\frac12)\) \(\approx\) \(0.1066920736 + 2.249234813i\)
\(L(1)\) \(\approx\) \(0.9733746926 + 1.105392873i\)
\(L(1)\) \(\approx\) \(0.9733746926 + 1.105392873i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2557 \( 1 \)
good2 \( 1 + (0.516 + 0.856i)T \)
3 \( 1 + (0.202 + 0.979i)T \)
5 \( 1 + (-0.0270 - 0.999i)T \)
7 \( 1 + (0.934 - 0.355i)T \)
11 \( 1 + (0.0172 + 0.999i)T \)
13 \( 1 + (0.163 + 0.986i)T \)
17 \( 1 + (0.905 + 0.423i)T \)
19 \( 1 + (0.836 + 0.548i)T \)
23 \( 1 + (-0.999 + 0.0245i)T \)
29 \( 1 + (0.998 + 0.0491i)T \)
31 \( 1 + (0.245 - 0.969i)T \)
37 \( 1 + (-0.756 - 0.654i)T \)
41 \( 1 + (0.997 + 0.0638i)T \)
43 \( 1 + (0.274 + 0.961i)T \)
47 \( 1 + (-0.163 + 0.986i)T \)
53 \( 1 + (-0.598 + 0.801i)T \)
59 \( 1 + (0.670 - 0.741i)T \)
61 \( 1 + (-0.883 + 0.467i)T \)
67 \( 1 + (0.998 - 0.0540i)T \)
71 \( 1 + (0.0761 + 0.997i)T \)
73 \( 1 + (-0.988 + 0.151i)T \)
79 \( 1 + (0.367 + 0.930i)T \)
83 \( 1 + (0.659 - 0.751i)T \)
89 \( 1 + (-0.307 - 0.951i)T \)
97 \( 1 + (-0.311 + 0.950i)T \)
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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

−19.17701865799397656576467519828, −18.29429405637291738760374253925, −18.07744595398568984500344713343, −17.45253383277018584235485997281, −15.97647026735129086483973917156, −15.2005711757690966807666857509, −14.44076539704694328493337318698, −13.84295662509813640795614766543, −13.650491937154669808331586940017, −12.355729076102819543309735861696, −11.94693121081454265721653534748, −11.2767034251308237218185334555, −10.639608844262441026700704334736, −9.86303175698104764834500008939, −8.71251221643045842775964705762, −8.13957576529438525971287904292, −7.31716267914258192142918629696, −6.33211872734800415499871234826, −5.65784366251507432278720997114, −5.05204417293132849087199806883, −3.59841624594276677658939693182, −3.06137040788905662930061275362, −2.44163395320884842845677303249, −1.468975824374253172778990122460, −0.614283226740866112759904214510, 1.29003195375039059969673160322, 2.3945976457681149595933455192, 3.7423432169552495349104259087, 4.249959109626891944356653827791, 4.755206109380332787607709035566, 5.486896563720161115863677368528, 6.22505160475578949120902147698, 7.624009789175606437188683828949, 7.891652519323330593881634937072, 8.73889969877929577172921426184, 9.502987580526064391197975798047, 10.04385636683402778815850207810, 11.297069839840292138516974425641, 11.98542429627705266103294062155, 12.55327676700036561239802311763, 13.67184842317051046511470112078, 14.288815196370343863131409631304, 14.59628111938371172806353546639, 15.68664746748189805278267398068, 16.06904751082411247815380531029, 16.748134682182107347386582303093, 17.392310305549254939824751891621, 17.872414856704189065358838697375, 19.081466898168795441777549452413, 20.18678747793944096715175507915

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