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Properties

Label 1-1045-1045.1009-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.995 + 0.0931i$
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  + (0.615 − 0.788i)2-s + (−0.719 − 0.694i)3-s + (−0.241 − 0.970i)4-s + (−0.990 + 0.139i)6-s + (0.913 − 0.406i)7-s + (−0.913 − 0.406i)8-s + (0.0348 + 0.999i)9-s + (−0.5 + 0.866i)12-s + (0.882 + 0.469i)13-s + (0.241 − 0.970i)14-s + (−0.882 + 0.469i)16-s + (0.0348 − 0.999i)17-s + (0.809 + 0.587i)18-s + (−0.939 − 0.342i)21-s + (−0.766 − 0.642i)23-s + (0.374 + 0.927i)24-s + ⋯
L(s)  = 1  + (0.615 − 0.788i)2-s + (−0.719 − 0.694i)3-s + (−0.241 − 0.970i)4-s + (−0.990 + 0.139i)6-s + (0.913 − 0.406i)7-s + (−0.913 − 0.406i)8-s + (0.0348 + 0.999i)9-s + (−0.5 + 0.866i)12-s + (0.882 + 0.469i)13-s + (0.241 − 0.970i)14-s + (−0.882 + 0.469i)16-s + (0.0348 − 0.999i)17-s + (0.809 + 0.587i)18-s + (−0.939 − 0.342i)21-s + (−0.766 − 0.642i)23-s + (0.374 + 0.927i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.995 + 0.0931i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.995 + 0.0931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06970318006 - 1.493727549i\)
\(L(\frac12)\) \(\approx\) \(-0.06970318006 - 1.493727549i\)
\(L(1)\) \(\approx\) \(0.7567130337 - 0.9059445577i\)
\(L(1)\) \(\approx\) \(0.7567130337 - 0.9059445577i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.615 - 0.788i)T \)
3 \( 1 + (-0.719 - 0.694i)T \)
7 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (0.882 + 0.469i)T \)
17 \( 1 + (0.0348 - 0.999i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (0.961 + 0.275i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.719 - 0.694i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.559 + 0.829i)T \)
53 \( 1 + (0.848 - 0.529i)T \)
59 \( 1 + (-0.559 - 0.829i)T \)
61 \( 1 + (0.374 - 0.927i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (-0.848 - 0.529i)T \)
73 \( 1 + (0.438 - 0.898i)T \)
79 \( 1 + (0.990 + 0.139i)T \)
83 \( 1 + (-0.978 - 0.207i)T \)
89 \( 1 + (-0.173 + 0.984i)T \)
97 \( 1 + (-0.615 + 0.788i)T \)
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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

−21.92403897016318549124429500866, −21.36489047128822839244024317658, −20.86864606005764041979260109790, −19.82733709175206045323115261069, −18.284353964271684518212300973552, −17.88777828904891027805587350961, −17.16208215068585656125655511816, −16.35187983923359947348182648259, −15.55797622715165105852971227931, −15.0992641866162029754483577896, −14.27522286902384427198771594862, −13.36399261124919184173301421043, −12.34845352563064895785332870021, −11.74478139621362321536645051552, −10.91636610637420692286617010463, −10.027409127126860143821211057837, −8.72767622297761318872911066716, −8.31729748515127377227798627324, −7.15706882780653674889677901422, −6.058545484524543571090536559716, −5.64060157147077860018990672495, −4.72027249295738270091838701073, −3.97629140082381558148254764333, −3.06643290463143593467473906561, −1.46784360909931749354760238456, 0.58672289346305069765453247549, 1.59662744629744069462182429283, 2.33926244797705497894179906788, 3.722991084804567669861630196270, 4.64327455923376521695970723527, 5.34378279088600317406549077363, 6.27160305878339696587329314746, 7.091369259214477924509469848, 8.1487717119597711289730562584, 9.15211282271656109846541109029, 10.41690673474663762656156513291, 10.90813415802664051571238542312, 11.713018679884250706554960377774, 12.201938081092739696304580693, 13.22968526882142891567988875001, 13.9239197148754483226318020365, 14.3734894903523578449954970026, 15.68219775515722614173980610875, 16.4340945361076254416262299953, 17.549797482473038451221508043240, 18.181234148621357309218233661119, 18.7340066827468864178329209554, 19.63361663641592671227727766077, 20.55074673322579264356306035260, 21.035491523234087911856698286318

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