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Properties

Label 1-1033-1033.129-r0-0-0
Degree $1$
Conductor $1033$
Sign $-0.849 + 0.527i$
Analytic cond. $4.79723$
Root an. cond. $4.79723$
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  + (0.520 − 0.853i)2-s + (0.391 + 0.920i)3-s + (−0.457 − 0.889i)4-s + (−0.934 + 0.357i)5-s + (0.989 + 0.145i)6-s + (0.520 + 0.853i)7-s + (−0.997 − 0.0729i)8-s + (−0.694 + 0.719i)9-s + (−0.181 + 0.983i)10-s + (−0.581 − 0.813i)11-s + (0.639 − 0.768i)12-s + (−0.694 + 0.719i)13-s + 14-s + (−0.694 − 0.719i)15-s + (−0.581 + 0.813i)16-s + (0.905 + 0.424i)17-s + ⋯
L(s)  = 1  + (0.520 − 0.853i)2-s + (0.391 + 0.920i)3-s + (−0.457 − 0.889i)4-s + (−0.934 + 0.357i)5-s + (0.989 + 0.145i)6-s + (0.520 + 0.853i)7-s + (−0.997 − 0.0729i)8-s + (−0.694 + 0.719i)9-s + (−0.181 + 0.983i)10-s + (−0.581 − 0.813i)11-s + (0.639 − 0.768i)12-s + (−0.694 + 0.719i)13-s + 14-s + (−0.694 − 0.719i)15-s + (−0.581 + 0.813i)16-s + (0.905 + 0.424i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.849 + 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1033\)
Sign: $-0.849 + 0.527i$
Analytic conductor: \(4.79723\)
Root analytic conductor: \(4.79723\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1033} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1033,\ (0:\ ),\ -0.849 + 0.527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1163066769 + 0.4079988334i\)
\(L(\frac12)\) \(\approx\) \(0.1163066769 + 0.4079988334i\)
\(L(1)\) \(\approx\) \(0.9506047580 + 0.006153932484i\)
\(L(1)\) \(\approx\) \(0.9506047580 + 0.006153932484i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1033 \( 1 \)
good2 \( 1 + (0.520 - 0.853i)T \)
3 \( 1 + (0.391 + 0.920i)T \)
5 \( 1 + (-0.934 + 0.357i)T \)
7 \( 1 + (0.520 + 0.853i)T \)
11 \( 1 + (-0.581 - 0.813i)T \)
13 \( 1 + (-0.694 + 0.719i)T \)
17 \( 1 + (0.905 + 0.424i)T \)
19 \( 1 + (0.109 - 0.994i)T \)
23 \( 1 + (0.989 + 0.145i)T \)
29 \( 1 + (-0.791 - 0.611i)T \)
31 \( 1 + (-0.934 + 0.357i)T \)
37 \( 1 + (-0.581 + 0.813i)T \)
41 \( 1 + (-0.694 - 0.719i)T \)
43 \( 1 + (-0.322 + 0.946i)T \)
47 \( 1 + (-0.997 - 0.0729i)T \)
53 \( 1 + (-0.934 + 0.357i)T \)
59 \( 1 + (-0.791 + 0.611i)T \)
61 \( 1 + (-0.791 - 0.611i)T \)
67 \( 1 + (0.252 + 0.967i)T \)
71 \( 1 + (-0.934 - 0.357i)T \)
73 \( 1 + (-0.581 - 0.813i)T \)
79 \( 1 + (-0.581 - 0.813i)T \)
83 \( 1 + (-0.997 + 0.0729i)T \)
89 \( 1 + (-0.0365 - 0.999i)T \)
97 \( 1 + (-0.0365 + 0.999i)T \)
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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

−20.97716595490829284443230234633, −20.50875679076708103169624671070, −19.86256955870997511686331955533, −18.697294902410983547541414981735, −18.14603451644686293860221657092, −17.11299188668759395797810937849, −16.70671503327288210910728932333, −15.58869969543886529317312899584, −14.68166784489517714404181359929, −14.45939623643823914775612398141, −13.24700583864009787638773432266, −12.67065806832117672171134297676, −12.130229481104219908750036861078, −11.13493520010030808893425171359, −9.82419876152632578135773905237, −8.693725679470908590905492019838, −7.76595039076790578120345488130, −7.5572184582793326990732896749, −6.92043254980787989569108487661, −5.4711923836651007365088448188, −4.87928134616749612132803329251, −3.72437658766968845160053931316, −3.04858848219212996415081271606, −1.55914656484885122761232212929, −0.136721936922287820565743724746, 1.7799762035167773262934333973, 2.99928605929544683644759837596, 3.24695943103812653249868169920, 4.5214099300849069950327643361, 5.02465776493970391649686366633, 5.949985998024391938360864483362, 7.41716510441716101333504728552, 8.4743980083138972941608280479, 9.066681722219426590530156403848, 10.01241963016272733614411385575, 10.99451016046643368719923514015, 11.36725364442063595266703126371, 12.13626113133842594606504690950, 13.18591101176441695999304141296, 14.18033417104955802868131175340, 14.84425861833942215311570108681, 15.29646322417656733961390259599, 16.112933437019918536114847138830, 17.146405549591274579337987834129, 18.46445021506656390798826805294, 19.05015572419283975186320056627, 19.53271193461615030932975062893, 20.52037244506054858714917766724, 21.21539265750108302292307678884, 21.78192383240302709973975299308

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