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Properties

Label 1-1573-1573.1022-r0-0-0
Degree $1$
Conductor $1573$
Sign $-0.326 - 0.945i$
Analytic cond. $7.30498$
Root an. cond. $7.30498$
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  + (−0.540 − 0.841i)2-s + 3-s + (−0.415 + 0.909i)4-s + (−0.989 − 0.142i)5-s + (−0.540 − 0.841i)6-s + (−0.755 − 0.654i)7-s + (0.989 − 0.142i)8-s + 9-s + (0.415 + 0.909i)10-s + (−0.415 + 0.909i)12-s + (−0.142 + 0.989i)14-s + (−0.989 − 0.142i)15-s + (−0.654 − 0.755i)16-s + (−0.959 − 0.281i)17-s + (−0.540 − 0.841i)18-s + (0.281 + 0.959i)19-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)2-s + 3-s + (−0.415 + 0.909i)4-s + (−0.989 − 0.142i)5-s + (−0.540 − 0.841i)6-s + (−0.755 − 0.654i)7-s + (0.989 − 0.142i)8-s + 9-s + (0.415 + 0.909i)10-s + (−0.415 + 0.909i)12-s + (−0.142 + 0.989i)14-s + (−0.989 − 0.142i)15-s + (−0.654 − 0.755i)16-s + (−0.959 − 0.281i)17-s + (−0.540 − 0.841i)18-s + (0.281 + 0.959i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1573\)    =    \(11^{2} \cdot 13\)
Sign: $-0.326 - 0.945i$
Analytic conductor: \(7.30498\)
Root analytic conductor: \(7.30498\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1573} (1022, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1573,\ (0:\ ),\ -0.326 - 0.945i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6250829497 - 0.8775716222i\)
\(L(\frac12)\) \(\approx\) \(0.6250829497 - 0.8775716222i\)
\(L(1)\) \(\approx\) \(0.7704525595 - 0.3958746165i\)
\(L(1)\) \(\approx\) \(0.7704525595 - 0.3958746165i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.540 - 0.841i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.989 - 0.142i)T \)
7 \( 1 + (-0.755 - 0.654i)T \)
17 \( 1 + (-0.959 - 0.281i)T \)
19 \( 1 + (0.281 + 0.959i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (0.959 - 0.281i)T \)
31 \( 1 + (-0.909 + 0.415i)T \)
37 \( 1 + (0.909 - 0.415i)T \)
41 \( 1 + (-0.540 - 0.841i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (0.540 - 0.841i)T \)
53 \( 1 + (-0.654 + 0.755i)T \)
59 \( 1 + (0.540 - 0.841i)T \)
61 \( 1 + (-0.841 - 0.540i)T \)
67 \( 1 + (0.540 + 0.841i)T \)
71 \( 1 + (-0.281 - 0.959i)T \)
73 \( 1 + (0.755 - 0.654i)T \)
79 \( 1 + (0.142 - 0.989i)T \)
83 \( 1 + (0.755 + 0.654i)T \)
89 \( 1 + (-0.281 + 0.959i)T \)
97 \( 1 + (0.989 - 0.142i)T \)
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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

−20.25890009755047035150891320045, −19.77777830540573576215571370818, −19.28956516704284721319277023506, −18.50492105790357532417067751225, −18.0448665480764288344601220904, −16.76888778801449979876367437812, −16.015554631937014239854894826, −15.50593255733928822884086255099, −14.97697271512879183714890620679, −14.30926392728555995934741058684, −13.18638343286195131486611016076, −12.79023143241173975242516736027, −11.50016474047576142750469561824, −10.65795837965972246407591300554, −9.68310806710676746479947609078, −8.98702554995018880468586228220, −8.48081106357633744841981162646, −7.691086957107118073064663280472, −6.845908675673932630545097140590, −6.359251644104156869443367026773, −4.915972122217835128037653513736, −4.27488421870529773570217417239, −3.13844050940363075959029936609, −2.37829760766276110996554619211, −0.94186481952937624639611042760, 0.539080354701279612477082146916, 1.65626296417804160597124877537, 2.74905843699996694950646064065, 3.57351167204178693132641683513, 3.95148780222675993401973567724, 4.926682630498815906391584985723, 6.744940079949291912491146239827, 7.39601086724070779981629816911, 8.02595696834806877495962264259, 8.908173527594092661064215237043, 9.42995272230898747555701764258, 10.37012613703862111985653833305, 10.97351400842290159712198910101, 12.05005882202580397448146183506, 12.62763367866562308394516885514, 13.43896542170055674840856839611, 13.9909869936471760756001826816, 15.13741623301023708342613648969, 15.91594526738463961275089960143, 16.46359407207387528169935920694, 17.41563650191548300143637875681, 18.44482220764039440333399395058, 19.03371853893807066516442170001, 19.62263197237999158057028755794, 20.221379910939238521880946288811

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