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Properties

Label 1-1045-1045.123-r0-0-0
Degree $1$
Conductor $1045$
Sign $0.951 + 0.307i$
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  + (−0.275 + 0.961i)2-s + (0.469 − 0.882i)3-s + (−0.848 − 0.529i)4-s + (0.719 + 0.694i)6-s + (0.743 + 0.669i)7-s + (0.743 − 0.669i)8-s + (−0.559 − 0.829i)9-s + (−0.866 + 0.5i)12-s + (−0.898 − 0.438i)13-s + (−0.848 + 0.529i)14-s + (0.438 + 0.898i)16-s + (0.829 + 0.559i)17-s + (0.951 − 0.309i)18-s + (0.939 − 0.342i)21-s + (0.642 + 0.766i)23-s + (−0.241 − 0.970i)24-s + ⋯
L(s)  = 1  + (−0.275 + 0.961i)2-s + (0.469 − 0.882i)3-s + (−0.848 − 0.529i)4-s + (0.719 + 0.694i)6-s + (0.743 + 0.669i)7-s + (0.743 − 0.669i)8-s + (−0.559 − 0.829i)9-s + (−0.866 + 0.5i)12-s + (−0.898 − 0.438i)13-s + (−0.848 + 0.529i)14-s + (0.438 + 0.898i)16-s + (0.829 + 0.559i)17-s + (0.951 − 0.309i)18-s + (0.939 − 0.342i)21-s + (0.642 + 0.766i)23-s + (−0.241 − 0.970i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.307i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1045\)    =    \(5 \cdot 11 \cdot 19\)
Sign: $0.951 + 0.307i$
Analytic conductor: \(4.85295\)
Root analytic conductor: \(4.85295\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1045} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1045,\ (0:\ ),\ 0.951 + 0.307i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.476735504 + 0.2329466078i\)
\(L(\frac12)\) \(\approx\) \(1.476735504 + 0.2329466078i\)
\(L(1)\) \(\approx\) \(1.076019428 + 0.1775688152i\)
\(L(1)\) \(\approx\) \(1.076019428 + 0.1775688152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.275 + 0.961i)T \)
3 \( 1 + (0.469 - 0.882i)T \)
7 \( 1 + (0.743 + 0.669i)T \)
13 \( 1 + (-0.898 - 0.438i)T \)
17 \( 1 + (0.829 + 0.559i)T \)
23 \( 1 + (0.642 + 0.766i)T \)
29 \( 1 + (0.0348 - 0.999i)T \)
31 \( 1 + (-0.104 + 0.994i)T \)
37 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (0.882 + 0.469i)T \)
43 \( 1 + (0.642 - 0.766i)T \)
47 \( 1 + (0.788 + 0.615i)T \)
53 \( 1 + (-0.0697 - 0.997i)T \)
59 \( 1 + (0.615 + 0.788i)T \)
61 \( 1 + (0.241 - 0.970i)T \)
67 \( 1 + (-0.342 + 0.939i)T \)
71 \( 1 + (-0.997 - 0.0697i)T \)
73 \( 1 + (-0.139 - 0.990i)T \)
79 \( 1 + (-0.719 + 0.694i)T \)
83 \( 1 + (0.406 - 0.913i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (0.275 - 0.961i)T \)
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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

−21.32256685603831763199113383462, −20.62621097225443305865367917984, −20.20354049143611182938224595787, −19.333239726879296812512765165, −18.61146203900050871131315225949, −17.59806128455760603756079896339, −16.79580799660216345920762502057, −16.36668948676571442997073913755, −14.886240603208597097960212985217, −14.39289110963292404627892293621, −13.7177067486271154578512299549, −12.70260202175040135055083041973, −11.71832426110514041541586876841, −11.00104392275826877443571804567, −10.32267091018609879443603389086, −9.56363923748016596180754577102, −8.8860132712671922649762370509, −7.90198567218904589425679711422, −7.28896242838698084474432015815, −5.45177154780493752815422410645, −4.61985806912863579886292735426, −4.0854524995308997583025453558, −2.978774648956337099242713124660, −2.23250871564408105948195356423, −0.93941274857456448189535157359, 0.908718607232420338105008922867, 1.94304487110330554437275669558, 3.06831811444402373041949234399, 4.38937401985921042456345373604, 5.49741677600339046957692857717, 5.99826640815050281224095884682, 7.24420030090064407866342602149, 7.69381479239849887116150155899, 8.463506538059536414180743759707, 9.20570311407628270237848295781, 10.08178846726194088684888797447, 11.31893434670497072683213781989, 12.322616187066341349001991459887, 12.949535586658015040012839321239, 13.93898693570762626188865227142, 14.677103489564928242383862717647, 15.03636063814750020153380943, 16.0300431466500362126720949255, 17.27105186053369478387572279851, 17.539364338074420907737144417450, 18.37756262417012672250896444779, 19.16863494284031980116854721888, 19.59435998789219971268849539340, 20.79084821651592612978364802309, 21.62865188191845543375231479736

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