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Properties

Label 1-1045-1045.214-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.315 - 0.948i$
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  + (0.241 − 0.970i)2-s + (−0.0348 + 0.999i)3-s + (−0.882 − 0.469i)4-s + (0.961 + 0.275i)6-s + (−0.669 − 0.743i)7-s + (−0.669 + 0.743i)8-s + (−0.997 − 0.0697i)9-s + (0.5 − 0.866i)12-s + (−0.559 + 0.829i)13-s + (−0.882 + 0.469i)14-s + (0.559 + 0.829i)16-s + (0.997 − 0.0697i)17-s + (−0.309 + 0.951i)18-s + (0.766 − 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.719 − 0.694i)24-s + ⋯
L(s)  = 1  + (0.241 − 0.970i)2-s + (−0.0348 + 0.999i)3-s + (−0.882 − 0.469i)4-s + (0.961 + 0.275i)6-s + (−0.669 − 0.743i)7-s + (−0.669 + 0.743i)8-s + (−0.997 − 0.0697i)9-s + (0.5 − 0.866i)12-s + (−0.559 + 0.829i)13-s + (−0.882 + 0.469i)14-s + (0.559 + 0.829i)16-s + (0.997 − 0.0697i)17-s + (−0.309 + 0.951i)18-s + (0.766 − 0.642i)21-s + (−0.173 + 0.984i)23-s + (−0.719 − 0.694i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.315 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.315 - 0.948i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.315 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5318643247 - 0.7372967722i\)
\(L(\frac12)\) \(\approx\) \(0.5318643247 - 0.7372967722i\)
\(L(1)\) \(\approx\) \(0.8246400211 - 0.2996225819i\)
\(L(1)\) \(\approx\) \(0.8246400211 - 0.2996225819i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.241 - 0.970i)T \)
3 \( 1 + (-0.0348 + 0.999i)T \)
7 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (-0.559 + 0.829i)T \)
17 \( 1 + (0.997 - 0.0697i)T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.848 - 0.529i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (0.0348 - 0.999i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.374 - 0.927i)T \)
53 \( 1 + (-0.438 - 0.898i)T \)
59 \( 1 + (-0.374 - 0.927i)T \)
61 \( 1 + (-0.719 + 0.694i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (0.438 - 0.898i)T \)
73 \( 1 + (0.615 - 0.788i)T \)
79 \( 1 + (0.961 - 0.275i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (0.241 - 0.970i)T \)
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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

−22.041559309911033045098393263971, −21.25378791850671898101815779812, −19.95987906195424725582595046901, −19.27038496269599829853697371651, −18.40423385820008953954495485119, −17.97477480436154430772045415298, −17.049353302839284429602540086977, −16.32647160956584267494482251314, −15.523077402762604285459159743916, −14.53204029900453163986826229811, −14.12174359647523370479046177975, −12.87888663144815023722912215103, −12.59032384236625857762890800354, −11.94666687745749933913502594497, −10.489837832580311000351432476439, −9.4599436749845367638042600656, −8.58132566861152788976917973087, −7.89838008495507058389207881873, −7.084873087787500187886296416429, −6.243243338391444651465819300690, −5.6544032533324767651547189753, −4.76646068378750691716871777429, −3.24867839527572816268210103740, −2.674721032594161882507656016585, −1.02553698506538371284223155667, 0.42220005254165842469046802386, 1.921291585952573682544140535356, 3.114452018077569070523017996399, 3.727449735060948522104243196245, 4.52291824922411907931087505799, 5.37439628455407510885836778200, 6.34086050200180112927564337190, 7.63671343543187950872656147165, 8.78130994128843473786961447083, 9.71036936379812661649956656793, 9.97232453883213070032920745214, 10.84901309861985137481860815603, 11.74585969234023079507448900758, 12.3208541132573502298336832774, 13.65291041331613506450952897934, 13.92910416812639995068780524190, 14.94602706374736823371042163307, 15.7319933960940064527517476133, 16.8835120980710241637295503642, 17.12052811413980745827496501510, 18.404479127919826904566294080667, 19.38831995890075822547772837779, 19.72046483089576830718796530899, 20.79302915648674361147324133089, 21.098470240405421937705548660564

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