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Properties

Label 1-1573-1573.424-r0-0-0
Degree $1$
Conductor $1573$
Sign $-0.998 - 0.0590i$
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  + (0.0570 + 0.998i)2-s + (0.309 + 0.951i)3-s + (−0.993 + 0.113i)4-s + (−0.884 − 0.466i)5-s + (−0.931 + 0.362i)6-s + (−0.389 − 0.921i)7-s + (−0.170 − 0.985i)8-s + (−0.809 + 0.587i)9-s + (0.415 − 0.909i)10-s + (−0.415 − 0.909i)12-s + (0.897 − 0.441i)14-s + (0.170 − 0.985i)15-s + (0.974 − 0.226i)16-s + (0.941 + 0.336i)17-s + (−0.633 − 0.774i)18-s + (−0.999 + 0.0285i)19-s + ⋯
L(s)  = 1  + (0.0570 + 0.998i)2-s + (0.309 + 0.951i)3-s + (−0.993 + 0.113i)4-s + (−0.884 − 0.466i)5-s + (−0.931 + 0.362i)6-s + (−0.389 − 0.921i)7-s + (−0.170 − 0.985i)8-s + (−0.809 + 0.587i)9-s + (0.415 − 0.909i)10-s + (−0.415 − 0.909i)12-s + (0.897 − 0.441i)14-s + (0.170 − 0.985i)15-s + (0.974 − 0.226i)16-s + (0.941 + 0.336i)17-s + (−0.633 − 0.774i)18-s + (−0.999 + 0.0285i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1573\)    =    \(11^{2} \cdot 13\)
Sign: $-0.998 - 0.0590i$
Analytic conductor: \(7.30498\)
Root analytic conductor: \(7.30498\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1573} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1573,\ (0:\ ),\ -0.998 - 0.0590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02093579580 + 0.7082410440i\)
\(L(\frac12)\) \(\approx\) \(0.02093579580 + 0.7082410440i\)
\(L(1)\) \(\approx\) \(0.5970786971 + 0.5043493972i\)
\(L(1)\) \(\approx\) \(0.5970786971 + 0.5043493972i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.0570 + 0.998i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.884 - 0.466i)T \)
7 \( 1 + (-0.389 - 0.921i)T \)
17 \( 1 + (0.941 + 0.336i)T \)
19 \( 1 + (-0.999 + 0.0285i)T \)
23 \( 1 + (0.654 - 0.755i)T \)
29 \( 1 + (0.564 - 0.825i)T \)
31 \( 1 + (-0.980 + 0.198i)T \)
37 \( 1 + (-0.676 + 0.736i)T \)
41 \( 1 + (0.967 + 0.254i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (0.633 - 0.774i)T \)
53 \( 1 + (0.974 + 0.226i)T \)
59 \( 1 + (-0.967 + 0.254i)T \)
61 \( 1 + (0.998 + 0.0570i)T \)
67 \( 1 + (-0.540 + 0.841i)T \)
71 \( 1 + (-0.791 + 0.610i)T \)
73 \( 1 + (-0.856 + 0.516i)T \)
79 \( 1 + (-0.696 - 0.717i)T \)
83 \( 1 + (0.996 - 0.0855i)T \)
89 \( 1 + (0.281 + 0.959i)T \)
97 \( 1 + (0.884 - 0.466i)T \)
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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

−19.88998582589450456260591603762, −19.21552615996782650113588929144, −18.96309847946741854135973880455, −18.25352131369458985189170624678, −17.55838806445873927321656415044, −16.47739195448890622248987054140, −15.36316955440286986221092494392, −14.67580490815345030847551179086, −14.06778267157059470199089997900, −13.025377685279012052446553500671, −12.42241904092790143652245737667, −11.98105967666108442163861255060, −11.17905829555377903626531278551, −10.420298872976319572283305585914, −9.16824940870390688676426483301, −8.77997667112877451766883789385, −7.8191194154439357196115746698, −7.09694698392246233710880909128, −5.99768494757418772553774444446, −5.20055706717450733988638105879, −3.88409693659339957777110497424, −3.17224245438030527722343201183, −2.52988129252840703202403938911, −1.578809761304314699742337170101, −0.31377125307564591099031172419, 0.95791647010564564608695888155, 2.948719191841121910900713095707, 3.86156808232174155139055464566, 4.28382524811426705094014556479, 5.06686849236628330891090798080, 6.0499431942981959917242209186, 7.06352223691395830938822545544, 7.845655727208986191595788253178, 8.502811685357813005802950153748, 9.18346527520890862510954894588, 10.15986416930197673564125096341, 10.68473969614388042056182163036, 11.87839236933622640023294908929, 12.84442221143727694326294999562, 13.47120172444640632576016730399, 14.57191705628159293923356809705, 14.83563300810981662912523794751, 15.7945923814054334726773041527, 16.36625265673884654224751509765, 16.8474260183340864523674467873, 17.4525726487088230532334948141, 18.87616067807250113934070628393, 19.32570814771101443280041361876, 20.1689051792188744169906462955, 20.9369817283448019640187436511

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