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Properties

Label 1-6043-6043.695-r0-0-0
Degree $1$
Conductor $6043$
Sign $0.00533 + 0.999i$
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  + (0.907 − 0.421i)2-s + (−0.998 − 0.0592i)3-s + (0.645 − 0.763i)4-s + (−0.944 − 0.329i)5-s + (−0.930 + 0.366i)6-s + (−0.689 + 0.724i)7-s + (0.263 − 0.964i)8-s + (0.992 + 0.118i)9-s + (−0.995 + 0.0986i)10-s + (−0.5 − 0.866i)11-s + (−0.689 + 0.724i)12-s + (−0.282 + 0.959i)13-s + (−0.320 + 0.947i)14-s + (0.922 + 0.384i)15-s + (−0.167 − 0.985i)16-s + (−0.465 − 0.885i)17-s + ⋯
L(s)  = 1  + (0.907 − 0.421i)2-s + (−0.998 − 0.0592i)3-s + (0.645 − 0.763i)4-s + (−0.944 − 0.329i)5-s + (−0.930 + 0.366i)6-s + (−0.689 + 0.724i)7-s + (0.263 − 0.964i)8-s + (0.992 + 0.118i)9-s + (−0.995 + 0.0986i)10-s + (−0.5 − 0.866i)11-s + (−0.689 + 0.724i)12-s + (−0.282 + 0.959i)13-s + (−0.320 + 0.947i)14-s + (0.922 + 0.384i)15-s + (−0.167 − 0.985i)16-s + (−0.465 − 0.885i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6043 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00533 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6043 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.00533 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6043\)
Sign: $0.00533 + 0.999i$
Analytic conductor: \(28.0635\)
Root analytic conductor: \(28.0635\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6043} (695, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6043,\ (0:\ ),\ 0.00533 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2856973597 - 0.2872266412i\)
\(L(\frac12)\) \(\approx\) \(-0.2856973597 - 0.2872266412i\)
\(L(1)\) \(\approx\) \(0.6898254245 - 0.4467770567i\)
\(L(1)\) \(\approx\) \(0.6898254245 - 0.4467770567i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad6043 \( 1 \)
good2 \( 1 + (0.907 - 0.421i)T \)
3 \( 1 + (-0.998 - 0.0592i)T \)
5 \( 1 + (-0.944 - 0.329i)T \)
7 \( 1 + (-0.689 + 0.724i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.282 + 0.959i)T \)
17 \( 1 + (-0.465 - 0.885i)T \)
19 \( 1 + (-0.660 - 0.751i)T \)
23 \( 1 + (-0.430 - 0.902i)T \)
29 \( 1 + (-0.598 - 0.800i)T \)
31 \( 1 + (-0.898 - 0.438i)T \)
37 \( 1 + (0.147 - 0.989i)T \)
41 \( 1 + (-0.984 + 0.176i)T \)
43 \( 1 + (0.782 - 0.622i)T \)
47 \( 1 + (0.996 + 0.0789i)T \)
53 \( 1 + (0.889 - 0.456i)T \)
59 \( 1 + (0.999 - 0.0395i)T \)
61 \( 1 + (0.447 + 0.894i)T \)
67 \( 1 + (0.263 - 0.964i)T \)
71 \( 1 + (-0.915 - 0.403i)T \)
73 \( 1 + (-0.0493 - 0.998i)T \)
79 \( 1 + (-0.357 + 0.933i)T \)
83 \( 1 + (0.992 - 0.118i)T \)
89 \( 1 + (-0.995 + 0.0986i)T \)
97 \( 1 + (0.614 + 0.788i)T \)
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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

−17.95944754124923517588890575989, −17.29913929105382028049936900047, −16.82780925417420624558352307494, −16.03807001421580873953123128847, −15.641464658963351863483375739269, −15.028007869568379100464869336148, −14.508098536627184929383887669074, −13.32010557631814490207170582792, −12.83729683471550365267114764658, −12.46473920694685084258895899152, −11.7653126247541006464606976285, −10.95887176013091105478740924330, −10.46151885358024037837114499837, −9.97491552544943536897178496022, −8.60522565840145260885841506730, −7.71679268941532026423946831182, −7.29533938428033485174602118266, −6.76616874736795207714221523944, −5.975872343101443112794468141799, −5.33727029776077428292242017680, −4.54453144935014041952987433057, −3.86887910833762086683935830506, −3.50177282393840421769925952492, −2.40758687012740531313537810639, −1.35081472857451076827965810038, 0.12837687814731857861150756761, 0.63906916065501970111806406763, 2.0848034958944105304537410602, 2.56453564337245180483923445770, 3.705814584590315258256745444789, 4.19600117464132580747976135509, 4.93525034801605496119030341818, 5.57091471182128953933510665556, 6.21106909409338184158157656785, 6.94272984271396550243831422165, 7.44148647987779614168002108663, 8.69312649674562091633395442144, 9.27777353939224381671111040032, 10.16118363471732946728498987513, 11.03358882929021333204085065447, 11.32473182641382423206982805029, 12.02935445163902496042978458509, 12.4261659543001355124206627661, 13.17991987062165776315967346875, 13.60914382479666878441924995640, 14.72354568091735703324142455055, 15.36172870124110284892046100519, 15.91668968626453107509381931833, 16.39803882307636983734587215948, 16.815984067983492386226017250973

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