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Properties

Label 1-6043-6043.160-r0-0-0
Degree $1$
Conductor $6043$
Sign $0.765 + 0.643i$
Analytic cond. $28.0635$
Root an. cond. $28.0635$
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  + (0.482 + 0.875i)2-s + (0.293 − 0.955i)3-s + (−0.533 + 0.845i)4-s + (−0.907 + 0.420i)5-s + (0.978 − 0.204i)6-s + (0.370 + 0.929i)7-s + (−0.998 − 0.0592i)8-s + (−0.827 − 0.561i)9-s + (−0.806 − 0.591i)10-s + (−0.879 − 0.475i)11-s + (0.651 + 0.758i)12-s + (−0.212 + 0.977i)13-s + (−0.634 + 0.772i)14-s + (0.135 + 0.990i)15-s + (−0.430 − 0.902i)16-s + (0.104 + 0.994i)17-s + ⋯
L(s)  = 1  + (0.482 + 0.875i)2-s + (0.293 − 0.955i)3-s + (−0.533 + 0.845i)4-s + (−0.907 + 0.420i)5-s + (0.978 − 0.204i)6-s + (0.370 + 0.929i)7-s + (−0.998 − 0.0592i)8-s + (−0.827 − 0.561i)9-s + (−0.806 − 0.591i)10-s + (−0.879 − 0.475i)11-s + (0.651 + 0.758i)12-s + (−0.212 + 0.977i)13-s + (−0.634 + 0.772i)14-s + (0.135 + 0.990i)15-s + (−0.430 − 0.902i)16-s + (0.104 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6043 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6043 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.765 + 0.643i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6043\)
Sign: $0.765 + 0.643i$
Analytic conductor: \(28.0635\)
Root analytic conductor: \(28.0635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6043} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6043,\ (0:\ ),\ 0.765 + 0.643i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9792178411 + 0.3572635284i\)
\(L(\frac12)\) \(\approx\) \(0.9792178411 + 0.3572635284i\)
\(L(1)\) \(\approx\) \(0.8745205818 + 0.3299751491i\)
\(L(1)\) \(\approx\) \(0.8745205818 + 0.3299751491i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6043 \( 1 \)
good2 \( 1 + (0.482 + 0.875i)T \)
3 \( 1 + (0.293 - 0.955i)T \)
5 \( 1 + (-0.907 + 0.420i)T \)
7 \( 1 + (0.370 + 0.929i)T \)
11 \( 1 + (-0.879 - 0.475i)T \)
13 \( 1 + (-0.212 + 0.977i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (-0.904 - 0.425i)T \)
23 \( 1 + (-0.126 - 0.992i)T \)
29 \( 1 + (-0.747 - 0.664i)T \)
31 \( 1 + (-0.549 - 0.835i)T \)
37 \( 1 + (0.992 + 0.124i)T \)
41 \( 1 + (-0.999 - 0.0155i)T \)
43 \( 1 + (0.122 - 0.992i)T \)
47 \( 1 + (0.287 + 0.957i)T \)
53 \( 1 + (-0.972 + 0.231i)T \)
59 \( 1 + (-0.528 - 0.848i)T \)
61 \( 1 + (0.504 + 0.863i)T \)
67 \( 1 + (0.849 + 0.527i)T \)
71 \( 1 + (0.562 - 0.826i)T \)
73 \( 1 + (-0.337 - 0.941i)T \)
79 \( 1 + (0.471 + 0.881i)T \)
83 \( 1 + (0.995 - 0.0996i)T \)
89 \( 1 + (0.990 + 0.136i)T \)
97 \( 1 + (0.239 - 0.970i)T \)
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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

−17.656129862568317077481591106596, −16.92031662033832607842917721254, −16.09644253155446339148265516609, −15.61562526819621862283696592279, −14.785237720069732369776022303, −14.59847924487076378704427497539, −13.53011413035541933425723766995, −13.08521594049112085146865865878, −12.35098162052507910467120016148, −11.567436855693804037768586721576, −10.88563581421273415857915102470, −10.57788598437609905723292957983, −9.78650460381628007229775360667, −9.245823057701605743419631046433, −8.22110704132209889086606707671, −7.85989974368574743743246976688, −6.972831638365221017517246127519, −5.60590698361270342855665217826, −4.99822711590592596095414334091, −4.68935266493879846544858271259, −3.742585681368771269989017084163, −3.427811729639213499275903298923, −2.57217190676117654755037913345, −1.59523657144781764782601872554, −0.49838364861370957594943422468, 0.39329233721889679359337437000, 2.05153279908632462576668518423, 2.49314424231294028566195994421, 3.370570812707403004142102990217, 4.14598132599266085793134435777, 4.87607741204575872430930005085, 5.891141033519916786703461722674, 6.28127612095742191870058749558, 6.99522612888134610852769453796, 7.817926292212151377028926669887, 8.15191016777699539900798956938, 8.715841122826430558784524571496, 9.41562962789706244114921564195, 10.81076408863087340261134672057, 11.39766450779257542501648864985, 12.06956731862725816129226314729, 12.64894036279998255680088535748, 13.13299837825925725851158167549, 13.996361637351851848132192813951, 14.62303797969085293052404068822, 15.09704002630296564758465644595, 15.527829137532125222759527345326, 16.49397670464000114980886330349, 17.00426497816136214812799510641, 17.82276022123106913673174855403

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