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Properties

Label 1-1045-1045.1014-r1-0-0
Degree $1$
Conductor $1045$
Sign $-0.725 + 0.687i$
Analytic cond. $112.300$
Root an. cond. $112.300$
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  + (0.913 − 0.406i)2-s + (−0.669 − 0.743i)3-s + (0.669 − 0.743i)4-s + (−0.913 − 0.406i)6-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s − 12-s + (−0.104 + 0.994i)13-s + (0.669 + 0.743i)14-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (0.309 + 0.951i)18-s + (0.5 − 0.866i)21-s + (0.5 + 0.866i)23-s + (−0.913 + 0.406i)24-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (−0.669 − 0.743i)3-s + (0.669 − 0.743i)4-s + (−0.913 − 0.406i)6-s + (0.309 + 0.951i)7-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s − 12-s + (−0.104 + 0.994i)13-s + (0.669 + 0.743i)14-s + (−0.104 − 0.994i)16-s + (−0.104 − 0.994i)17-s + (0.309 + 0.951i)18-s + (0.5 − 0.866i)21-s + (0.5 + 0.866i)23-s + (−0.913 + 0.406i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.725 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.725 + 0.687i$
Analytic conductor: \(112.300\)
Root analytic conductor: \(112.300\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1014, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (1:\ ),\ -0.725 + 0.687i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1608816735 - 0.4037107913i\)
\(L(\frac12)\) \(\approx\) \(-0.1608816735 - 0.4037107913i\)
\(L(1)\) \(\approx\) \(1.136546303 - 0.5118871420i\)
\(L(1)\) \(\approx\) \(1.136546303 - 0.5118871420i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.913 - 0.406i)T \)
3 \( 1 + (-0.669 - 0.743i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.104 + 0.994i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.669 + 0.743i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (-0.669 - 0.743i)T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (0.978 - 0.207i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (-0.913 - 0.406i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.104 - 0.994i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.913 + 0.406i)T \)
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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

−21.92621667353914750367575763933, −21.27137529147584374094521964598, −20.30190309403072191330298790758, −20.13456832672747007491570901305, −18.55022794028933926427783882691, −17.37663958986738809672818866864, −17.08919849784535633037337220479, −16.38659472535002683858245485554, −15.32071746379966741712720109851, −14.998604739493826717712081471816, −14.048266679108025665826017235717, −13.114946046330598298660844932299, −12.42990653981477867606232609990, −11.500574843214201584401101589876, −10.64362169300142161147882925716, −10.28591211128422454391594442543, −8.828766242088891242066021733996, −7.88788298149695813917783933738, −6.97523602182556272491941965436, −6.123871136412854998668008470228, −5.313582521909192624290538348084, −4.50833885822962292494728172418, −3.81492498894457302129105079444, −2.943256095637208770330256815991, −1.37408647535306573463821767616, 0.06494830309596767293340870924, 1.515018477089691012259593799799, 2.10624391965465059758900844179, 3.147766758698479473728617495645, 4.45338831363510835402680142403, 5.30729125457826777660845308899, 5.81144498848704870085349860745, 6.90139276997984287439701577943, 7.42810858895936799783994982221, 8.85435609093464594212716128174, 9.70252888336127154892409261733, 11.05147495104420688430726955242, 11.39901692306802761194170419252, 12.14850298490735272540127401038, 12.79369980191447914523662037626, 13.67128566489168812641035142885, 14.33465685421672136053682979662, 15.26596436924095824553958550262, 16.11279603995367856847488705169, 16.84446298101372814740154532225, 18.004993094762508089386935728527, 18.64703314931537357862375524834, 19.218021064144542895547067754062, 20.134814796105442682042108952727, 21.09451863676438962778992696719

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