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Properties

Label 1-1573-1573.1486-r0-0-0
Degree $1$
Conductor $1573$
Sign $-0.514 + 0.857i$
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  + (−0.723 − 0.690i)2-s + (−0.5 − 0.866i)3-s + (0.0475 + 0.998i)4-s + (0.654 + 0.755i)5-s + (−0.235 + 0.971i)6-s + (0.995 + 0.0950i)7-s + (0.654 − 0.755i)8-s + (−0.5 + 0.866i)9-s + (0.0475 − 0.998i)10-s + (0.841 − 0.540i)12-s + (−0.654 − 0.755i)14-s + (0.327 − 0.945i)15-s + (−0.995 + 0.0950i)16-s + (−0.786 − 0.618i)17-s + (0.959 − 0.281i)18-s + (−0.928 − 0.371i)19-s + ⋯
L(s)  = 1  + (−0.723 − 0.690i)2-s + (−0.5 − 0.866i)3-s + (0.0475 + 0.998i)4-s + (0.654 + 0.755i)5-s + (−0.235 + 0.971i)6-s + (0.995 + 0.0950i)7-s + (0.654 − 0.755i)8-s + (−0.5 + 0.866i)9-s + (0.0475 − 0.998i)10-s + (0.841 − 0.540i)12-s + (−0.654 − 0.755i)14-s + (0.327 − 0.945i)15-s + (−0.995 + 0.0950i)16-s + (−0.786 − 0.618i)17-s + (0.959 − 0.281i)18-s + (−0.928 − 0.371i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1573\)    =    \(11^{2} \cdot 13\)
Sign: $-0.514 + 0.857i$
Analytic conductor: \(7.30498\)
Root analytic conductor: \(7.30498\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1573} (1486, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1573,\ (0:\ ),\ -0.514 + 0.857i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03534686179 + 0.06245333527i\)
\(L(\frac12)\) \(\approx\) \(0.03534686179 + 0.06245333527i\)
\(L(1)\) \(\approx\) \(0.5607097836 - 0.1983205734i\)
\(L(1)\) \(\approx\) \(0.5607097836 - 0.1983205734i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.723 - 0.690i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.654 + 0.755i)T \)
7 \( 1 + (0.995 + 0.0950i)T \)
17 \( 1 + (-0.786 - 0.618i)T \)
19 \( 1 + (-0.928 - 0.371i)T \)
23 \( 1 + (-0.995 + 0.0950i)T \)
29 \( 1 + (-0.786 + 0.618i)T \)
31 \( 1 + (-0.841 - 0.540i)T \)
37 \( 1 + (-0.0475 + 0.998i)T \)
41 \( 1 + (-0.723 - 0.690i)T \)
43 \( 1 + (-0.327 - 0.945i)T \)
47 \( 1 + (0.959 + 0.281i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.723 + 0.690i)T \)
61 \( 1 + (0.723 - 0.690i)T \)
67 \( 1 + (-0.723 - 0.690i)T \)
71 \( 1 + (-0.928 - 0.371i)T \)
73 \( 1 + (-0.415 + 0.909i)T \)
79 \( 1 + (-0.654 - 0.755i)T \)
83 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (-0.928 + 0.371i)T \)
97 \( 1 + (0.327 + 0.945i)T \)
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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

−20.28288243647183243651501987928, −19.646268035805516811734084476092, −18.35103087000802439994714904446, −17.76414769596422097690018079057, −17.248647952737006885014069402723, −16.61765268372878987504161414340, −16.02123966456256127655921430695, −15.049077203316057294142302562612, −14.60709519172639820505613685937, −13.72434161542819585893831809893, −12.69499955007554747659582568083, −11.61594712592176044445161384872, −10.877414523692922847035608610963, −10.22594333486556833199214150418, −9.50175014346725100111184285352, −8.61852150263563278704934359340, −8.279585230636806801743406067054, −7.01619869676452032135453224760, −5.94585497039145581020646397174, −5.598514664942550826718510383819, −4.59295438840039496948203874842, −4.10183952646644716427663093888, −2.17023011789533350084591791602, −1.44227749485485274852260080172, −0.03422682755210127847111228189, 1.48019305750618168604826420581, 2.06200961353799628931446553988, 2.72166188182503896159946674873, 4.058620856679038843231615928014, 5.14483302066316044961843857642, 6.11471623014092131750868157559, 7.04427805464242704828773919204, 7.53461736769459720014027599718, 8.522999703163867058100055989986, 9.20411713593091555935074866639, 10.44792932659736392954214608073, 10.81508372979737686028304439248, 11.57993832837860455528158745206, 12.16212367584873613325232632983, 13.252751899230678979959091684477, 13.667613099038354604070001271566, 14.61053754867077787835052686160, 15.61125964729852627450256439306, 16.825923020772921025028770472104, 17.31336923399440697606454022018, 17.907531889253875242109742289080, 18.57563920261497618634304395497, 18.855330978405595356221216986830, 20.08602984649690157374248027009, 20.491657943051490886488644396

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