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Properties

Label 1-1045-1045.764-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.388 + 0.921i$
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  + (0.997 + 0.0697i)2-s + (0.615 + 0.788i)3-s + (0.990 + 0.139i)4-s + (0.559 + 0.829i)6-s + (0.978 − 0.207i)7-s + (0.978 + 0.207i)8-s + (−0.241 + 0.970i)9-s + (0.5 + 0.866i)12-s + (−0.961 + 0.275i)13-s + (0.990 − 0.139i)14-s + (0.961 + 0.275i)16-s + (0.241 + 0.970i)17-s + (−0.309 + 0.951i)18-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (0.438 + 0.898i)24-s + ⋯
L(s)  = 1  + (0.997 + 0.0697i)2-s + (0.615 + 0.788i)3-s + (0.990 + 0.139i)4-s + (0.559 + 0.829i)6-s + (0.978 − 0.207i)7-s + (0.978 + 0.207i)8-s + (−0.241 + 0.970i)9-s + (0.5 + 0.866i)12-s + (−0.961 + 0.275i)13-s + (0.990 − 0.139i)14-s + (0.961 + 0.275i)16-s + (0.241 + 0.970i)17-s + (−0.309 + 0.951i)18-s + (0.766 + 0.642i)21-s + (−0.173 − 0.984i)23-s + (0.438 + 0.898i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.388 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.388 + 0.921i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (764, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.388 + 0.921i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.309956759 + 2.196848200i\)
\(L(\frac12)\) \(\approx\) \(3.309956759 + 2.196848200i\)
\(L(1)\) \(\approx\) \(2.354674007 + 0.8811677405i\)
\(L(1)\) \(\approx\) \(2.354674007 + 0.8811677405i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.997 + 0.0697i)T \)
3 \( 1 + (0.615 + 0.788i)T \)
7 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (-0.961 + 0.275i)T \)
17 \( 1 + (0.241 + 0.970i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.374 - 0.927i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.615 - 0.788i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (-0.848 - 0.529i)T \)
53 \( 1 + (0.719 + 0.694i)T \)
59 \( 1 + (0.848 - 0.529i)T \)
61 \( 1 + (0.438 - 0.898i)T \)
67 \( 1 + (-0.766 + 0.642i)T \)
71 \( 1 + (-0.719 + 0.694i)T \)
73 \( 1 + (-0.0348 - 0.999i)T \)
79 \( 1 + (0.559 - 0.829i)T \)
83 \( 1 + (0.104 - 0.994i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (0.997 + 0.0697i)T \)
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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

−21.230530050762585527344004060371, −20.75860478842350881908425219709, −19.89447592938886897540295868315, −19.3602257091000547022653851706, −18.30946058176703567582809355953, −17.62018599200173621487752260211, −16.653876256615876173304647268895, −15.5175953382165698508411453240, −14.847761807170703213239465483951, −14.26125507780235844915330021481, −13.58210310247902558978478873309, −12.78304592816625654645677584690, −11.8608298636530928410958536718, −11.57303741357069447417344270362, −10.30712657780359825063934427590, −9.28912704924091814337261417507, −8.14437075324892039833071334524, −7.460283271954378126017853801808, −6.84039009963797674048865487152, −5.600229290989252499877447085376, −4.99933667719896923494904261470, −3.85340845888266275428990408472, −2.84800293759793983909706438500, −2.12841673452183534183504106463, −1.17505894255386535689676692818, 1.6960277178289302401411090014, 2.478318177451544736478839597809, 3.50471370003800893657388744174, 4.46242672427718047083865213437, 4.84252418065526537088021329269, 5.891812094418188940348238125040, 7.033599573810547457987467376024, 7.98283213397950702316428005484, 8.53540068429135371504749466060, 9.97426029955073416529960434713, 10.47250826082140796203344483661, 11.44992499069835167460302884633, 12.13995375509172285832755055940, 13.23699188457412031569749000239, 13.972788683223664934362511387861, 14.77670281873558060143006843916, 14.976667789679886358051015647698, 16.0104630335442267937158457779, 16.87377445825945275816439411177, 17.39828467186085359482097975612, 18.918926901004559478603933839874, 19.62381197137995978368707037115, 20.45946094763951442488763637786, 20.9745404974375461060512340988, 21.6628152754633458628850924813

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