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Properties

Label 1-1045-1045.358-r0-0-0
Degree $1$
Conductor $1045$
Sign $-0.999 + 0.00716i$
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  + (0.694 − 0.719i)2-s + (−0.529 + 0.848i)3-s + (−0.0348 − 0.999i)4-s + (0.241 + 0.970i)6-s + (−0.743 + 0.669i)7-s + (−0.743 − 0.669i)8-s + (−0.438 − 0.898i)9-s + (0.866 + 0.5i)12-s + (−0.0697 + 0.997i)13-s + (−0.0348 + 0.999i)14-s + (−0.997 + 0.0697i)16-s + (0.898 + 0.438i)17-s + (−0.951 − 0.309i)18-s + (−0.173 − 0.984i)21-s + (−0.342 − 0.939i)23-s + (0.961 − 0.275i)24-s + ⋯
L(s)  = 1  + (0.694 − 0.719i)2-s + (−0.529 + 0.848i)3-s + (−0.0348 − 0.999i)4-s + (0.241 + 0.970i)6-s + (−0.743 + 0.669i)7-s + (−0.743 − 0.669i)8-s + (−0.438 − 0.898i)9-s + (0.866 + 0.5i)12-s + (−0.0697 + 0.997i)13-s + (−0.0348 + 0.999i)14-s + (−0.997 + 0.0697i)16-s + (0.898 + 0.438i)17-s + (−0.951 − 0.309i)18-s + (−0.173 − 0.984i)21-s + (−0.342 − 0.939i)23-s + (0.961 − 0.275i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00716i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $-0.999 + 0.00716i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (358, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ -0.999 + 0.00716i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0008357692100 - 0.2333550919i\)
\(L(\frac12)\) \(\approx\) \(0.0008357692100 - 0.2333550919i\)
\(L(1)\) \(\approx\) \(0.8594312708 - 0.1711652058i\)
\(L(1)\) \(\approx\) \(0.8594312708 - 0.1711652058i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.694 - 0.719i)T \)
3 \( 1 + (-0.529 + 0.848i)T \)
7 \( 1 + (-0.743 + 0.669i)T \)
13 \( 1 + (-0.0697 + 0.997i)T \)
17 \( 1 + (0.898 + 0.438i)T \)
23 \( 1 + (-0.342 - 0.939i)T \)
29 \( 1 + (-0.882 - 0.469i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (-0.848 - 0.529i)T \)
43 \( 1 + (-0.342 + 0.939i)T \)
47 \( 1 + (-0.139 - 0.990i)T \)
53 \( 1 + (0.829 + 0.559i)T \)
59 \( 1 + (-0.990 - 0.139i)T \)
61 \( 1 + (-0.961 - 0.275i)T \)
67 \( 1 + (-0.984 - 0.173i)T \)
71 \( 1 + (0.559 + 0.829i)T \)
73 \( 1 + (-0.927 + 0.374i)T \)
79 \( 1 + (-0.241 + 0.970i)T \)
83 \( 1 + (-0.406 - 0.913i)T \)
89 \( 1 + (-0.766 - 0.642i)T \)
97 \( 1 + (-0.694 + 0.719i)T \)
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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

−22.467859927376351445712606813381, −21.37092781053772595961064608121, −20.336044096456128529539594726814, −19.66247530642100734865142108937, −18.627202057366458345940682821300, −17.8472411486922986904173393934, −17.12208953966540076362273708718, −16.49665161438757426014582409966, −15.78392448339554639905388660902, −14.79782976115811795501223094386, −13.75984909070930571429223750132, −13.41768959512339161850782408239, −12.49159889401386887237523347181, −12.027997305756092560993739587084, −10.927964369089463479512836978071, −10.01852307460330989738631391894, −8.76709022192187786959600395534, −7.669627017551127934965167545225, −7.3145930472193862509480749378, −6.41030074501413971166039010831, −5.60250727566416834082213427275, −4.95167540107334480924158374712, −3.562618092880712265677564580377, −2.95220907242292985000790710589, −1.43763972529636134194536945598, 0.08028443422018832152027163904, 1.73654921678115221793246753201, 2.84750940055452473422994524020, 3.736279532793560771048795310511, 4.4253842693756472745391105706, 5.49744697816021365072678887102, 6.02717630644011828703609957224, 6.898514337785105534470172516599, 8.61291121931867189946654038099, 9.44222117093908602121765786027, 10.00671490195762741478574508943, 10.81414864816919345127593454551, 11.789526136042270105310210198891, 12.164818974550393335959330154677, 13.07777652882991766019872530359, 14.11043801295093336605813126800, 14.8950611949787185014011299360, 15.49774037804571308009612903942, 16.40691625643462939238380402826, 16.993007879856201588872054604111, 18.41450183911835095936833568846, 18.80812083117323345559039409199, 19.77472449802947965213261484780, 20.60794217444178940391936202948, 21.37224150813832585037269064494

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