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Properties

Label 1-2057-2057.213-r0-0-0
Degree $1$
Conductor $2057$
Sign $-0.227 - 0.973i$
Analytic cond. $9.55266$
Root an. cond. $9.55266$
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  + (−0.825 + 0.564i)2-s + (−0.156 − 0.987i)3-s + (0.362 − 0.931i)4-s + (0.926 + 0.376i)5-s + (0.686 + 0.727i)6-s + (0.963 − 0.268i)7-s + (0.226 + 0.974i)8-s + (−0.951 + 0.309i)9-s + (−0.977 + 0.212i)10-s + (−0.977 − 0.212i)12-s + (−0.774 − 0.633i)13-s + (−0.644 + 0.765i)14-s + (0.226 − 0.974i)15-s + (−0.736 − 0.676i)16-s + (0.610 − 0.791i)18-s + (0.884 + 0.466i)19-s + ⋯
L(s)  = 1  + (−0.825 + 0.564i)2-s + (−0.156 − 0.987i)3-s + (0.362 − 0.931i)4-s + (0.926 + 0.376i)5-s + (0.686 + 0.727i)6-s + (0.963 − 0.268i)7-s + (0.226 + 0.974i)8-s + (−0.951 + 0.309i)9-s + (−0.977 + 0.212i)10-s + (−0.977 − 0.212i)12-s + (−0.774 − 0.633i)13-s + (−0.644 + 0.765i)14-s + (0.226 − 0.974i)15-s + (−0.736 − 0.676i)16-s + (0.610 − 0.791i)18-s + (0.884 + 0.466i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2057 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2057\)    =    \(11^{2} \cdot 17\)
Sign: $-0.227 - 0.973i$
Analytic conductor: \(9.55266\)
Root analytic conductor: \(9.55266\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2057} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2057,\ (0:\ ),\ -0.227 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6193716098 - 0.7811301654i\)
\(L(\frac12)\) \(\approx\) \(0.6193716098 - 0.7811301654i\)
\(L(1)\) \(\approx\) \(0.7873148014 - 0.1819980799i\)
\(L(1)\) \(\approx\) \(0.7873148014 - 0.1819980799i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
17 \( 1 \)
good2 \( 1 + (-0.825 + 0.564i)T \)
3 \( 1 + (-0.156 - 0.987i)T \)
5 \( 1 + (0.926 + 0.376i)T \)
7 \( 1 + (0.963 - 0.268i)T \)
13 \( 1 + (-0.774 - 0.633i)T \)
19 \( 1 + (0.884 + 0.466i)T \)
23 \( 1 + (-0.349 - 0.936i)T \)
29 \( 1 + (0.0142 - 0.999i)T \)
31 \( 1 + (-0.863 + 0.504i)T \)
37 \( 1 + (-0.746 - 0.665i)T \)
41 \( 1 + (-0.903 + 0.428i)T \)
43 \( 1 + (-0.755 - 0.654i)T \)
47 \( 1 + (-0.610 - 0.791i)T \)
53 \( 1 + (0.676 + 0.736i)T \)
59 \( 1 + (0.336 + 0.941i)T \)
61 \( 1 + (-0.184 - 0.982i)T \)
67 \( 1 + (-0.959 + 0.281i)T \)
71 \( 1 + (-0.576 - 0.817i)T \)
73 \( 1 + (0.552 - 0.833i)T \)
79 \( 1 + (-0.240 - 0.970i)T \)
83 \( 1 + (-0.113 - 0.993i)T \)
89 \( 1 + (0.142 - 0.989i)T \)
97 \( 1 + (0.376 + 0.926i)T \)
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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

−20.21233697075079615727502707711, −19.58181596412594429463084924822, −18.40144144132048112369253994525, −17.85598528199305535105217207999, −17.254016050203063409913338512281, −16.678927786057235119236081201393, −15.99627558829971705490910988345, −15.09414767319679636006242849948, −14.28383702903859896870077336856, −13.55411444730137696026401381817, −12.48770616274763391774405892854, −11.63978016192245962890552115661, −11.27642443791484124168864376161, −10.24775345454962090045436414664, −9.74833580699666465722058295878, −9.075522494165107001202034732397, −8.53365306742782749604895715620, −7.53667780627677336389511798394, −6.57432572949462969516530244586, −5.33669799576237753159996201384, −4.96013587872868917729987867451, −3.90331080584532096189972680623, −2.915558945961803998154200250049, −2.022532326806408712709321648620, −1.24906517142874062653934720064, 0.44776562384053236233653762856, 1.6284033370365181268188155267, 2.02912341345817917417218410782, 3.06854118407209771528760526144, 4.86640561696917665324017729263, 5.49216591844418533479787088068, 6.15190443771499641847800872581, 7.09835637314162074655095373806, 7.52276175205446465769719689845, 8.33182283721178067273738437513, 9.05490648843651159929649112594, 10.20240810544392268466483470582, 10.50006765647216739854608995920, 11.53905152591292748835851252497, 12.170353168086156386654345109696, 13.34201033767424839392133832487, 13.9592447941752376589065868688, 14.57130868285156797191862127523, 15.121068937975040730539009870853, 16.44513548994938517822196380526, 17.02215830836648436142664775382, 17.61615136413934935683790213804, 18.21526375100007955196698908301, 18.51103517715416737710801043487, 19.54067232548551895207623028088

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