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Properties

Label 1-1573-1573.1010-r0-0-0
Degree $1$
Conductor $1573$
Sign $0.780 - 0.624i$
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  + (−0.969 − 0.244i)2-s + (−0.978 + 0.207i)3-s + (0.879 + 0.475i)4-s + (−0.870 − 0.491i)5-s + (0.999 + 0.0380i)6-s + (−0.935 + 0.353i)7-s + (−0.736 − 0.676i)8-s + (0.913 − 0.406i)9-s + (0.723 + 0.690i)10-s + (−0.959 − 0.281i)12-s + (0.993 − 0.113i)14-s + (0.953 + 0.299i)15-s + (0.548 + 0.836i)16-s + (0.820 + 0.572i)17-s + (−0.985 + 0.170i)18-s + (0.123 − 0.992i)19-s + ⋯
L(s)  = 1  + (−0.969 − 0.244i)2-s + (−0.978 + 0.207i)3-s + (0.879 + 0.475i)4-s + (−0.870 − 0.491i)5-s + (0.999 + 0.0380i)6-s + (−0.935 + 0.353i)7-s + (−0.736 − 0.676i)8-s + (0.913 − 0.406i)9-s + (0.723 + 0.690i)10-s + (−0.959 − 0.281i)12-s + (0.993 − 0.113i)14-s + (0.953 + 0.299i)15-s + (0.548 + 0.836i)16-s + (0.820 + 0.572i)17-s + (−0.985 + 0.170i)18-s + (0.123 − 0.992i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1573\)    =    \(11^{2} \cdot 13\)
Sign: $0.780 - 0.624i$
Analytic conductor: \(7.30498\)
Root analytic conductor: \(7.30498\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1573} (1010, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1573,\ (0:\ ),\ 0.780 - 0.624i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3411415785 - 0.1197106333i\)
\(L(\frac12)\) \(\approx\) \(0.3411415785 - 0.1197106333i\)
\(L(1)\) \(\approx\) \(0.3949155555 - 0.04093184184i\)
\(L(1)\) \(\approx\) \(0.3949155555 - 0.04093184184i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.969 - 0.244i)T \)
3 \( 1 + (-0.978 + 0.207i)T \)
5 \( 1 + (-0.870 - 0.491i)T \)
7 \( 1 + (-0.935 + 0.353i)T \)
17 \( 1 + (0.820 + 0.572i)T \)
19 \( 1 + (0.123 - 0.992i)T \)
23 \( 1 + (0.0475 - 0.998i)T \)
29 \( 1 + (-0.999 - 0.0190i)T \)
31 \( 1 + (0.941 + 0.336i)T \)
37 \( 1 + (-0.432 + 0.901i)T \)
41 \( 1 + (-0.0665 - 0.997i)T \)
43 \( 1 + (0.580 + 0.814i)T \)
47 \( 1 + (-0.985 - 0.170i)T \)
53 \( 1 + (-0.998 - 0.0570i)T \)
59 \( 1 + (-0.0665 + 0.997i)T \)
61 \( 1 + (-0.969 + 0.244i)T \)
67 \( 1 + (0.928 - 0.371i)T \)
71 \( 1 + (-0.683 + 0.730i)T \)
73 \( 1 + (-0.254 + 0.967i)T \)
79 \( 1 + (0.198 - 0.980i)T \)
83 \( 1 + (-0.362 + 0.931i)T \)
89 \( 1 + (0.981 + 0.189i)T \)
97 \( 1 + (0.00951 - 0.999i)T \)
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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

−20.37159927465170565870389370497, −19.50030764336555437301497929858, −18.889404240976815738792904448285, −18.54649469291409515368391951607, −17.56029877930695318216765509588, −16.86292735358912511668767628447, −16.13818172157491282677644243306, −15.81784062425488261607035788665, −14.87925294007521512037636989315, −13.90635849956412838737487561152, −12.73066943362031943341016335908, −12.000922420993542621673359999965, −11.40326540883012701500144018264, −10.62487261727674582298171307869, −9.96047011754462515697147456008, −9.29672602369699231903444735201, −7.77197320638879592965243725443, −7.62799929422830807070127730602, −6.665088290907107784538217703624, −6.07430713473228197792678828594, −5.16090452209094695493698701097, −3.83390361300608881223964577248, −3.03737210272129912982863712875, −1.66159001227933182912227522495, −0.56887725931340380625314818595, 0.43257284000387779230318120009, 1.39773177510802436155941249384, 2.84753279492937436628572049518, 3.67185648965316893859352345844, 4.64796304628260758593948460689, 5.72763131589864739714039825126, 6.56595975955178089820633059937, 7.234490900607220085397376389744, 8.19802951751051871477759532724, 9.01773028260768538006121559250, 9.757627338428083567566218707987, 10.51368730771184334202719747376, 11.27253172551863277743194411250, 12.01983452874308366683451376015, 12.51290721173087470803205562190, 13.166047028163956973744565090189, 14.90342185104089081069647659981, 15.60185274286285343050651452074, 16.10086179780381991745197617921, 16.77381140040196832844399249182, 17.287004647045988805952347327326, 18.288471499472269543540365865125, 18.99616794397452319448924086027, 19.413229829759607854696800964238, 20.37402394355084378002924747445

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