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Properties

Label 1-1045-1045.1004-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.492 - 0.870i$
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  + (−0.990 − 0.139i)2-s + (0.241 − 0.970i)3-s + (0.961 + 0.275i)4-s + (−0.374 + 0.927i)6-s + (−0.913 + 0.406i)7-s + (−0.913 − 0.406i)8-s + (−0.882 − 0.469i)9-s + (0.5 − 0.866i)12-s + (−0.848 + 0.529i)13-s + (0.961 − 0.275i)14-s + (0.848 + 0.529i)16-s + (0.882 − 0.469i)17-s + (0.809 + 0.587i)18-s + (0.173 + 0.984i)21-s + (0.939 − 0.342i)23-s + (−0.615 + 0.788i)24-s + ⋯
L(s)  = 1  + (−0.990 − 0.139i)2-s + (0.241 − 0.970i)3-s + (0.961 + 0.275i)4-s + (−0.374 + 0.927i)6-s + (−0.913 + 0.406i)7-s + (−0.913 − 0.406i)8-s + (−0.882 − 0.469i)9-s + (0.5 − 0.866i)12-s + (−0.848 + 0.529i)13-s + (0.961 − 0.275i)14-s + (0.848 + 0.529i)16-s + (0.882 − 0.469i)17-s + (0.809 + 0.587i)18-s + (0.173 + 0.984i)21-s + (0.939 − 0.342i)23-s + (−0.615 + 0.788i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.492 - 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.492 - 0.870i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (1004, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.492 - 0.870i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6790837039 - 0.3959818415i\)
\(L(\frac12)\) \(\approx\) \(0.6790837039 - 0.3959818415i\)
\(L(1)\) \(\approx\) \(0.6358099340 - 0.2115357588i\)
\(L(1)\) \(\approx\) \(0.6358099340 - 0.2115357588i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.990 - 0.139i)T \)
3 \( 1 + (0.241 - 0.970i)T \)
7 \( 1 + (-0.913 + 0.406i)T \)
13 \( 1 + (-0.848 + 0.529i)T \)
17 \( 1 + (0.882 - 0.469i)T \)
23 \( 1 + (0.939 - 0.342i)T \)
29 \( 1 + (-0.719 + 0.694i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.241 + 0.970i)T \)
43 \( 1 + (0.939 + 0.342i)T \)
47 \( 1 + (-0.438 - 0.898i)T \)
53 \( 1 + (-0.0348 - 0.999i)T \)
59 \( 1 + (0.438 - 0.898i)T \)
61 \( 1 + (-0.615 - 0.788i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (0.0348 - 0.999i)T \)
73 \( 1 + (0.997 - 0.0697i)T \)
79 \( 1 + (-0.374 - 0.927i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.990 - 0.139i)T \)
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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

−21.4097865780704287301673634738, −20.78334798733481306141121463140, −19.98849693937564381247992854710, −19.35944351523988893180118363569, −18.80706940024383374475592483025, −17.42205176356673191578234063552, −16.98954607938084045983937579289, −16.35323108698948358770320728599, −15.45033792241242471052932716329, −14.99033532322020019325136454187, −14.04309126769631786259156880632, −12.88779222462834024020073228234, −11.94003626683972398715353313672, −10.902895869366473556542242368916, −10.28989134477597911811771883227, −9.591617187805317455686029610774, −9.09634904661572858030086628347, −7.90851257107731242683114937247, −7.36212090267023842034234894092, −6.10891561068389387618038574187, −5.43490280133764128362400134778, −4.11543490623213200345543889952, −3.137599682629620341657373183868, −2.40378665941308458354416275087, −0.75605719779032937669459687979, 0.66253252589321318784115195792, 1.80002251656850231848368886020, 2.77283894884242162311151472673, 3.338005931854667809565292266264, 5.16479152612154577115535321771, 6.309662145121727891791974560969, 6.8725315023820551115607835640, 7.628455258665366305854104637786, 8.51716349517588145385184589488, 9.34934912861042745615804400470, 9.884901801154626176174295170520, 11.109912867447914298584499241338, 11.95687715818572095822855249985, 12.50085708117739547796885566157, 13.26591752525395082598608247097, 14.466793375384281751627581521237, 15.10483704773249578005851157585, 16.322555635606922981977271452225, 16.74991229737571251195750421324, 17.68883979342583541145922517150, 18.50292986010570210965139097546, 19.03065750206976643760547609109, 19.5614353124827215584582986160, 20.32313127400709343508471552939, 21.198683669663835282195177816829

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