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Properties

Label 1-2527-2527.1068-r0-0-0
Degree $1$
Conductor $2527$
Sign $0.0255 - 0.999i$
Analytic cond. $11.7353$
Root an. cond. $11.7353$
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  + (0.870 − 0.492i)2-s + (0.957 − 0.289i)3-s + (0.515 − 0.856i)4-s + (0.999 + 0.0367i)5-s + (0.690 − 0.723i)6-s + (0.0275 − 0.999i)8-s + (0.832 − 0.554i)9-s + (0.888 − 0.459i)10-s + (−0.401 + 0.915i)11-s + (0.245 − 0.969i)12-s + (−0.800 + 0.599i)13-s + (0.967 − 0.254i)15-s + (−0.467 − 0.883i)16-s + (0.484 − 0.875i)17-s + (0.451 − 0.892i)18-s + ⋯
L(s)  = 1  + (0.870 − 0.492i)2-s + (0.957 − 0.289i)3-s + (0.515 − 0.856i)4-s + (0.999 + 0.0367i)5-s + (0.690 − 0.723i)6-s + (0.0275 − 0.999i)8-s + (0.832 − 0.554i)9-s + (0.888 − 0.459i)10-s + (−0.401 + 0.915i)11-s + (0.245 − 0.969i)12-s + (−0.800 + 0.599i)13-s + (0.967 − 0.254i)15-s + (−0.467 − 0.883i)16-s + (0.484 − 0.875i)17-s + (0.451 − 0.892i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0255 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2527 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0255 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2527\)    =    \(7 \cdot 19^{2}\)
Sign: $0.0255 - 0.999i$
Analytic conductor: \(11.7353\)
Root analytic conductor: \(11.7353\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2527} (1068, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2527,\ (0:\ ),\ 0.0255 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.609841626 - 3.518651081i\)
\(L(\frac12)\) \(\approx\) \(3.609841626 - 3.518651081i\)
\(L(1)\) \(\approx\) \(2.434916440 - 1.279318936i\)
\(L(1)\) \(\approx\) \(2.434916440 - 1.279318936i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.870 - 0.492i)T \)
3 \( 1 + (0.957 - 0.289i)T \)
5 \( 1 + (0.999 + 0.0367i)T \)
11 \( 1 + (-0.401 + 0.915i)T \)
13 \( 1 + (-0.800 + 0.599i)T \)
17 \( 1 + (0.484 - 0.875i)T \)
23 \( 1 + (-0.227 - 0.973i)T \)
29 \( 1 + (0.933 - 0.359i)T \)
31 \( 1 + (0.635 - 0.771i)T \)
37 \( 1 + (-0.592 - 0.805i)T \)
41 \( 1 + (0.418 + 0.908i)T \)
43 \( 1 + (-0.842 + 0.539i)T \)
47 \( 1 + (-0.971 - 0.236i)T \)
53 \( 1 + (0.280 + 0.959i)T \)
59 \( 1 + (-0.995 - 0.0917i)T \)
61 \( 1 + (0.663 + 0.748i)T \)
67 \( 1 + (0.989 - 0.146i)T \)
71 \( 1 + (0.315 + 0.948i)T \)
73 \( 1 + (0.515 + 0.856i)T \)
79 \( 1 + (-0.842 + 0.539i)T \)
83 \( 1 + (0.137 - 0.990i)T \)
89 \( 1 + (0.515 - 0.856i)T \)
97 \( 1 + (-0.367 - 0.929i)T \)
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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

−19.7163355579777583745550580617, −19.10572802218923124593494711197, −18.0067323192648363710996152812, −17.33656081774178043985344345806, −16.623614841976201482176445250933, −15.844924249865191058904885874313, −15.23297249440591078793383493297, −14.46734642365791319127503829394, −13.93690047532461584817259731163, −13.405384587225225545692112612721, −12.73588685531039938266278008178, −12.00944074431668910374960981753, −10.72971050249013849482398598542, −10.2418408067613169249692295777, −9.35926287171834723047487881556, −8.34937703849531578969195484168, −8.05952641342989377983380405367, −6.989080652845061720453602359794, −6.25041394523148050016181996045, −5.25151605249431102143485534682, −4.9581283638163046757793006858, −3.62658524665509647576983422513, −3.145322285846115904636573040121, −2.35128763178343840299601098531, −1.47517011639556694649666725174, 1.01831387556400798739409306608, 2.02984497913098876397915415791, 2.46813331737475307244135272906, 3.12148488359441804715377604016, 4.426794979935865648764050583593, 4.76973633430367740699294842571, 5.860302542187629731348222594281, 6.75055581153457577732842315340, 7.22313213602651725540111839410, 8.291864658053013595205699463257, 9.429320529745006662111766088014, 9.82630242430057046556656607696, 10.328082997271732228776181557234, 11.55959801707130761884636102192, 12.34295521214137344451153496274, 12.86146806428976543783788050987, 13.57682029102110134004927725506, 14.28482885379207282553826083496, 14.55374267026291912513968694438, 15.413496786238150952555921045506, 16.20339912744233581793675144795, 17.17744932536500238785868567834, 18.20076301659250892780440107664, 18.57136332263431799004513237294, 19.47170329172257952857011517889

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