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Properties

Label 1-1037-1037.100-r0-0-0
Degree $1$
Conductor $1037$
Sign $0.869 - 0.493i$
Analytic cond. $4.81580$
Root an. cond. $4.81580$
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  + (0.994 + 0.104i)2-s + (0.987 − 0.156i)3-s + (0.978 + 0.207i)4-s + (−0.0523 − 0.998i)5-s + (0.998 − 0.0523i)6-s + (−0.358 − 0.933i)7-s + (0.951 + 0.309i)8-s + (0.951 − 0.309i)9-s + (0.0523 − 0.998i)10-s + (0.707 + 0.707i)11-s + (0.998 + 0.0523i)12-s + (0.5 + 0.866i)13-s + (−0.258 − 0.965i)14-s + (−0.207 − 0.978i)15-s + (0.913 + 0.406i)16-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.987 − 0.156i)3-s + (0.978 + 0.207i)4-s + (−0.0523 − 0.998i)5-s + (0.998 − 0.0523i)6-s + (−0.358 − 0.933i)7-s + (0.951 + 0.309i)8-s + (0.951 − 0.309i)9-s + (0.0523 − 0.998i)10-s + (0.707 + 0.707i)11-s + (0.998 + 0.0523i)12-s + (0.5 + 0.866i)13-s + (−0.258 − 0.965i)14-s + (−0.207 − 0.978i)15-s + (0.913 + 0.406i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1037 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1037\)    =    \(17 \cdot 61\)
Sign: $0.869 - 0.493i$
Analytic conductor: \(4.81580\)
Root analytic conductor: \(4.81580\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1037} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1037,\ (0:\ ),\ 0.869 - 0.493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.211046251 - 1.112345943i\)
\(L(\frac12)\) \(\approx\) \(4.211046251 - 1.112345943i\)
\(L(1)\) \(\approx\) \(2.684664380 - 0.4081264189i\)
\(L(1)\) \(\approx\) \(2.684664380 - 0.4081264189i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.994 + 0.104i)T \)
3 \( 1 + (0.987 - 0.156i)T \)
5 \( 1 + (-0.0523 - 0.998i)T \)
7 \( 1 + (-0.358 - 0.933i)T \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.406 + 0.913i)T \)
23 \( 1 + (-0.453 + 0.891i)T \)
29 \( 1 + (-0.258 - 0.965i)T \)
31 \( 1 + (-0.777 - 0.629i)T \)
37 \( 1 + (-0.156 + 0.987i)T \)
41 \( 1 + (-0.987 - 0.156i)T \)
43 \( 1 + (-0.207 - 0.978i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.951 - 0.309i)T \)
59 \( 1 + (-0.994 - 0.104i)T \)
67 \( 1 + (-0.669 + 0.743i)T \)
71 \( 1 + (0.998 + 0.0523i)T \)
73 \( 1 + (0.998 + 0.0523i)T \)
79 \( 1 + (-0.838 - 0.544i)T \)
83 \( 1 + (-0.994 - 0.104i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (-0.777 - 0.629i)T \)
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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

−21.90959876132909532812793399301, −21.02301058169935839432091521548, −19.95234245674299898798569743412, −19.62052590861853482065207422976, −18.65593757364158884861712644301, −18.0993953871778037637155972896, −16.5068677289241970863053527278, −15.684237746659063551506897044196, −15.27343503864877272656515083190, −14.36232432701685084124425272201, −14.01713399249355288734594410801, −12.984716418584544550742915539856, −12.34785031620215144116073397568, −11.1964400050624960520882739455, −10.666609496239164516029834369894, −9.60131476866941142075583178409, −8.708369346164835422091157380143, −7.72520561583038176340756106311, −6.77202282134234909897343049297, −6.10106622674809396594001042980, −5.09484938429252703995554421888, −3.84918124586051220117599779067, −3.13811371425287254778170442341, −2.707274691160002043422966810228, −1.57310976896554408883513519649, 1.429721778698226778575639467760, 1.90039383168130536843122651709, 3.52022329646793418010953286023, 3.93185379921178368516772717513, 4.64569266609762724193372390300, 5.896756366185265204675801926015, 6.88798011302148625291411283643, 7.559464082552552962252543710618, 8.40033960208234076471517499623, 9.50360164205269218173075767063, 10.08027434487008846502920141756, 11.51441900906897845827805511521, 12.1838623215503243292193701034, 13.00798029015946050629248247667, 13.72106756330885375981194652715, 14.07103232050053585985354269664, 15.15590999952035578377858559963, 15.81878184032830438186251701360, 16.70738833189971305767519633154, 17.18322269758048911504158094473, 18.67900826596109194522287267035, 19.554774793594080713029017908978, 20.29408539489702957219757171292, 20.5017051509122962507554225445, 21.34202822980670133171184751053

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