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Properties

Label 1-1045-1045.64-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.367 - 0.929i$
Analytic cond. $4.85295$
Root an. cond. $4.85295$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.367 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.367 - 0.929i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.367 - 0.929i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3254316131 - 0.4786222764i\)
\(L(\frac12)\) \(\approx\) \(0.3254316131 - 0.4786222764i\)
\(L(1)\) \(\approx\) \(0.5327251155 - 0.1619627366i\)
\(L(1)\) \(\approx\) \(0.5327251155 - 0.1619627366i\)

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

−21.71692011411586553662891291349, −20.990113780155602927501486417757, −20.34826508677395020934649032067, −19.30553521537878615689684431938, −18.521861402315949758635795172576, −18.04715572794761692247371898868, −17.05357324789781469043474660294, −16.33614763194464735537599154546, −15.88240154712851921725779425818, −15.04661332217673286475707625305, −13.98217263241020971419713276778, −12.44111361076345214238306911089, −12.2232105024843880433985208496, −11.2161675498999287632310256017, −10.69831993547985242114562087518, −9.55743134728509365012865044558, −9.20829413360428488319378367884, −8.42316516222185761837660673985, −7.0287568841398752801103923572, −6.43621095317912116965317369150, −5.355299541234550903082678856787, −4.36527849997990242781696045198, −3.245925538158762600393008403369, −2.42496410169269953010714995370, −1.06135274452310523229652074087, 0.45439393155846958877973368519, 1.316715171158199425275962285745, 2.424829742177262428284824260305, 3.795715851683234527285099731015, 5.2191536813369509407031887730, 5.947399458241295926572679125216, 6.7286640961441604374714410136, 7.595729918084366170668911734601, 8.007020823100449489079182308580, 9.200539006312648934121342281910, 10.29214113494351269269778482618, 10.7066396031508703452714786847, 11.53178690306137284604794290772, 12.60428152670888658692868295857, 13.31730925326901109711456340409, 14.2282266139100252082371777095, 15.25082787118605323855034410797, 16.07321900662639357184139115986, 16.882182120774116394363595231590, 17.440473185891822393751473844006, 17.93367502047533401350943314055, 18.98447002408466519528044255283, 19.50536930299862772498951948215, 20.16426970118919753702460790821, 21.1498559043222966173766910241

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