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Properties

Label 1-1045-1045.733-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.414 - 0.910i$
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  + (0.743 + 0.669i)2-s + (−0.994 − 0.104i)3-s + (0.104 + 0.994i)4-s + (−0.669 − 0.743i)6-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (0.978 + 0.207i)9-s − i·12-s + (−0.207 + 0.978i)13-s + (0.104 − 0.994i)14-s + (−0.978 + 0.207i)16-s + (−0.207 − 0.978i)17-s + (0.587 + 0.809i)18-s + (0.5 + 0.866i)21-s + (−0.866 − 0.5i)23-s + (0.669 − 0.743i)24-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (−0.994 − 0.104i)3-s + (0.104 + 0.994i)4-s + (−0.669 − 0.743i)6-s + (−0.587 − 0.809i)7-s + (−0.587 + 0.809i)8-s + (0.978 + 0.207i)9-s − i·12-s + (−0.207 + 0.978i)13-s + (0.104 − 0.994i)14-s + (−0.978 + 0.207i)16-s + (−0.207 − 0.978i)17-s + (0.587 + 0.809i)18-s + (0.5 + 0.866i)21-s + (−0.866 − 0.5i)23-s + (0.669 − 0.743i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.414 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.414 - 0.910i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (733, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.414 - 0.910i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5355043315 - 0.3444866836i\)
\(L(\frac12)\) \(\approx\) \(0.5355043315 - 0.3444866836i\)
\(L(1)\) \(\approx\) \(0.8524141852 + 0.2160536541i\)
\(L(1)\) \(\approx\) \(0.8524141852 + 0.2160536541i\)

Euler product

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

−21.7956756061730130385873772720, −21.27313273207454224670137065045, −20.09796666037401266186230579417, −19.52341169977959727371422601722, −18.49344763490107844552982092355, −18.049712244808597001410581515217, −16.98072979439815384760749057605, −16.0058993120816360357007135831, −15.3587826653404179123104143808, −14.758424581098458003740268738295, −13.43949832244797436952038710497, −12.741458894091929989591316778335, −12.27829059651450025923390884467, −11.44038618532879125558413040913, −10.64196555980172991417434720648, −9.95309897398520222420585203568, −9.18019069238458183256659011267, −7.805711372407946657270367803490, −6.52822713105120926270032993488, −5.91176659912916920081442452499, −5.33008310439398715479304717432, −4.304293122252154266434964784821, −3.406334987066808993781564284250, −2.354632240743795609290827325092, −1.228203346352671030175447441, 0.243571977106771240618282521118, 1.93390046604650754460214045688, 3.27408939830769686786344069374, 4.32931736692868851218163069121, 4.803580545621437204169470269, 5.9472420539458017798689109848, 6.68314770607815939201176371815, 7.14153427868930862053681775523, 8.14488058069987317071599296999, 9.43284343622511309790963798245, 10.25215564141531040126934222285, 11.375383718788420994726296007128, 11.90822371203681545799833319688, 12.75051765229340781463810715588, 13.59874522978914176146511716416, 14.0871935560020450949526271435, 15.28524070251304310334283968737, 16.15414901503578683436452727584, 16.51687162200873863165625440469, 17.249049733914533749614198353918, 18.03105879228346342098975991571, 18.9021045653437627981587313449, 19.995895309908485978763938487480, 20.86674424195163537916965558769, 21.720573861977699878133244241650

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