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Properties

Label 1-1045-1045.453-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.999 - 0.0368i$
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  + (0.898 + 0.438i)2-s + (0.927 − 0.374i)3-s + (0.615 + 0.788i)4-s + (0.997 + 0.0697i)6-s + (0.207 − 0.978i)7-s + (0.207 + 0.978i)8-s + (0.719 − 0.694i)9-s + (0.866 + 0.5i)12-s + (−0.970 + 0.241i)13-s + (0.615 − 0.788i)14-s + (−0.241 + 0.970i)16-s + (0.694 − 0.719i)17-s + (0.951 − 0.309i)18-s + (−0.173 − 0.984i)21-s + (−0.342 − 0.939i)23-s + (0.559 + 0.829i)24-s + ⋯
L(s)  = 1  + (0.898 + 0.438i)2-s + (0.927 − 0.374i)3-s + (0.615 + 0.788i)4-s + (0.997 + 0.0697i)6-s + (0.207 − 0.978i)7-s + (0.207 + 0.978i)8-s + (0.719 − 0.694i)9-s + (0.866 + 0.5i)12-s + (−0.970 + 0.241i)13-s + (0.615 − 0.788i)14-s + (−0.241 + 0.970i)16-s + (0.694 − 0.719i)17-s + (0.951 − 0.309i)18-s + (−0.173 − 0.984i)21-s + (−0.342 − 0.939i)23-s + (0.559 + 0.829i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.999 - 0.0368i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (453, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.999 - 0.0368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.808888441 - 0.07010738877i\)
\(L(\frac12)\) \(\approx\) \(3.808888441 - 0.07010738877i\)
\(L(1)\) \(\approx\) \(2.424215642 + 0.1230070070i\)
\(L(1)\) \(\approx\) \(2.424215642 + 0.1230070070i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.898 + 0.438i)T \)
3 \( 1 + (0.927 - 0.374i)T \)
7 \( 1 + (0.207 - 0.978i)T \)
13 \( 1 + (-0.970 + 0.241i)T \)
17 \( 1 + (0.694 - 0.719i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (0.990 - 0.139i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.374 + 0.927i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (-0.469 + 0.882i)T \)
53 \( 1 + (-0.275 + 0.961i)T \)
59 \( 1 + (0.882 - 0.469i)T \)
61 \( 1 + (-0.559 + 0.829i)T \)
67 \( 1 + (-0.984 - 0.173i)T \)
71 \( 1 + (0.961 - 0.275i)T \)
73 \( 1 + (0.529 - 0.848i)T \)
79 \( 1 + (-0.997 + 0.0697i)T \)
83 \( 1 + (-0.994 + 0.104i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (-0.898 - 0.438i)T \)
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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

−21.5941846936099331467186196032, −20.97359753076713280540150584278, −20.03212279933426883643131965559, −19.4063503157112941619330709549, −18.87825736309573206512477592778, −17.79772773707589900557196054800, −16.571409394302863292457569500863, −15.5728213794121122192410253186, −15.19021496749055843962874128400, −14.42106960890958130494276972660, −13.81590730126943301951842901021, −12.78500981219298620658481692887, −12.21277465713034855415758883577, −11.35715197873198109924649881020, −10.149386271564518846841254487981, −9.81278520515086282413726739998, −8.66926127545346752523248549161, −7.8437185373961423002811567797, −6.804409537843910870152023356199, −5.62104167081188059805667616415, −4.9851961835751334209467280031, −4.01195296254769442929240887700, −3.07915029311127475990611274103, −2.39817525767783632124868258050, −1.499941513876707197307814769283, 1.16765731536585101323606842683, 2.49520443991870117464551795527, 3.07980230770998537526622922181, 4.3173360488559905027189452251, 4.67403524110435513781971235557, 6.169402593523451582159525240024, 6.919865658436042919671044354952, 7.72261284937527303617410632698, 8.152079768921867351771421177493, 9.46121965556151687569652286802, 10.26877297659264072042972463809, 11.4801750827483083513215252014, 12.28235876880408906423319060694, 13.02457610360443678487460706701, 13.85044970795105409418946312923, 14.35797066042210667008259434254, 14.89818358225145246091065083287, 15.984098506374642579803553355551, 16.65796415363611536081482754315, 17.52804238811816650332532325729, 18.40042511295549268116498234395, 19.60633691983302731870761966503, 19.98099676200870787997697432612, 20.93328716252894209830712971221, 21.31351136166522264222679459039

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