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Properties

Label 1-1045-1045.259-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.432 - 0.901i$
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  + (0.104 + 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (0.104 − 0.994i)6-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + 12-s + (−0.913 − 0.406i)13-s + (0.978 + 0.207i)14-s + (0.913 − 0.406i)16-s + (0.913 − 0.406i)17-s + (−0.309 + 0.951i)18-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (0.104 + 0.994i)24-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.978 − 0.207i)3-s + (−0.978 + 0.207i)4-s + (0.104 − 0.994i)6-s + (0.309 − 0.951i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + 12-s + (−0.913 − 0.406i)13-s + (0.978 + 0.207i)14-s + (0.913 − 0.406i)16-s + (0.913 − 0.406i)17-s + (−0.309 + 0.951i)18-s + (−0.5 + 0.866i)21-s + (0.5 + 0.866i)23-s + (0.104 + 0.994i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.432 - 0.901i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.432 - 0.901i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5120047815 - 0.3221092707i\)
\(L(\frac12)\) \(\approx\) \(0.5120047815 - 0.3221092707i\)
\(L(1)\) \(\approx\) \(0.6795456171 + 0.1254185201i\)
\(L(1)\) \(\approx\) \(0.6795456171 + 0.1254185201i\)

Euler product

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

−21.53505475700495879487765408601, −21.22984667582145810874292517221, −20.27773242765676800889747334605, −19.12632587021853401746608256103, −18.681719900327630411057983945502, −17.95545155479479624554074175251, −17.06545249495646444666507139738, −16.519705475979492365308712528025, −15.078737562758798519076731232233, −14.813797995636095391881243336510, −13.529489601337505016702949851785, −12.61806116451025980880852046928, −11.92940619486180523639592739460, −11.62787920179973270138514265008, −10.48496351753303583035816186812, −9.95872483397158169608921515962, −9.05340380351589587994202887023, −8.157817952325781611281207013331, −6.86848460385374940217230749011, −5.77197553908262892272204731140, −5.10859518199979529842085444765, −4.41410864462266546167559620628, −3.26150701655098541857671668641, −2.1907953303049480066166730263, −1.209045686345069635549175318830, 0.32545451861702065105489909112, 1.44518561402949714959037128936, 3.27828905026939132742605070351, 4.36445180582601309473890649108, 5.083378611001915886981713788750, 5.77206059767798697846440051951, 6.81400978491832281720507012949, 7.48633925538084320322392535532, 7.96464609006538801779199353678, 9.49666295879672179598221164852, 10.04187741048437502888432851264, 11.05355416675014526450997910602, 11.97280922363701600429640460009, 12.85307611444102575785587074314, 13.50575283331688907590396598435, 14.40379346254443114672241279433, 15.185791220211436132218873135515, 16.11422519192078520541146117859, 16.897513566678883276150571815583, 17.21556621755432298522955619384, 18.00537338059595819248664668916, 18.79167585510288113909986812922, 19.656908854737351506939033194505, 20.85469014995132588017482078309, 21.66531530032982686393715045249

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