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Properties

Label 1-71e2-5041.409-r0-0-0
Degree $1$
Conductor $5041$
Sign $0.516 + 0.856i$
Analytic cond. $23.4103$
Root an. cond. $23.4103$
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  + (0.502 − 0.864i)2-s + (−0.540 + 0.841i)3-s + (−0.494 − 0.868i)4-s + (0.375 − 0.926i)5-s + (0.455 + 0.889i)6-s + (0.471 + 0.881i)7-s + (−0.999 − 0.00884i)8-s + (−0.416 − 0.909i)9-s + (−0.612 − 0.790i)10-s + (0.309 + 0.951i)11-s + (0.998 + 0.0530i)12-s + (0.502 + 0.864i)13-s + (0.999 + 0.0353i)14-s + (0.576 + 0.816i)15-s + (−0.510 + 0.860i)16-s + (−0.180 + 0.983i)17-s + ⋯
L(s)  = 1  + (0.502 − 0.864i)2-s + (−0.540 + 0.841i)3-s + (−0.494 − 0.868i)4-s + (0.375 − 0.926i)5-s + (0.455 + 0.889i)6-s + (0.471 + 0.881i)7-s + (−0.999 − 0.00884i)8-s + (−0.416 − 0.909i)9-s + (−0.612 − 0.790i)10-s + (0.309 + 0.951i)11-s + (0.998 + 0.0530i)12-s + (0.502 + 0.864i)13-s + (0.999 + 0.0353i)14-s + (0.576 + 0.816i)15-s + (−0.510 + 0.860i)16-s + (−0.180 + 0.983i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5041\)    =    \(71^{2}\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(23.4103\)
Root analytic conductor: \(23.4103\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5041} (409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5041,\ (0:\ ),\ 0.516 + 0.856i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.234335533 + 0.6969691029i\)
\(L(\frac12)\) \(\approx\) \(1.234335533 + 0.6969691029i\)
\(L(1)\) \(\approx\) \(1.101300209 - 0.1457992099i\)
\(L(1)\) \(\approx\) \(1.101300209 - 0.1457992099i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.502 - 0.864i)T \)
3 \( 1 + (-0.540 + 0.841i)T \)
5 \( 1 + (0.375 - 0.926i)T \)
7 \( 1 + (0.471 + 0.881i)T \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.502 + 0.864i)T \)
17 \( 1 + (-0.180 + 0.983i)T \)
19 \( 1 + (-0.875 - 0.483i)T \)
23 \( 1 + (0.814 - 0.580i)T \)
29 \( 1 + (-0.416 + 0.909i)T \)
31 \( 1 + (-0.742 + 0.670i)T \)
37 \( 1 + (0.814 - 0.580i)T \)
41 \( 1 + (-0.0221 - 0.999i)T \)
43 \( 1 + (0.292 - 0.956i)T \)
47 \( 1 + (-0.266 + 0.963i)T \)
53 \( 1 + (0.206 - 0.978i)T \)
59 \( 1 + (0.843 - 0.536i)T \)
61 \( 1 + (0.895 - 0.444i)T \)
67 \( 1 + (-0.692 + 0.721i)T \)
73 \( 1 + (-0.892 + 0.452i)T \)
79 \( 1 + (0.591 + 0.806i)T \)
83 \( 1 + (0.782 + 0.622i)T \)
89 \( 1 + (0.0132 + 0.999i)T \)
97 \( 1 + (-0.999 - 0.0442i)T \)
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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

−17.8327862075311236842098444426, −17.23994041001942659441634752268, −16.68225894033298531919464286426, −16.125145279264810161612979828664, −14.9296238995282066769116320497, −14.71585303637369939795538308395, −13.728234723618357629246284394042, −13.38044252726313146728271529395, −13.05520230024512244935655087459, −11.77736730824205440247715727429, −11.32396967828213373701855254896, −10.78314097857014647064798645070, −9.862881941246686879061939953647, −8.83680048345445895441818329818, −7.9529416575185504294229656614, −7.57947993820561735301596277477, −6.89484663584330237571998827597, −6.15796648910498072040343140176, −5.82176971003172298477110422037, −4.966891812967905871168830935444, −4.04640355147317106413938202274, −3.22168036742969944657338165818, −2.53149521124030850117262072128, −1.34954068844269137919451570087, −0.36261582041419894942898753141, 1.07408412778773477385789146828, 1.86986992753437126503796203588, 2.4481953381614380690049971058, 3.8130760117886158939378981140, 4.1707220361728752510134794087, 4.999358712608259229875171375, 5.32192941437123205027222056955, 6.17581920790289097547001775282, 6.81389237329633823626414165270, 8.49822248482864998091832118345, 8.93098708632822307655878210624, 9.30522033312761888168522294312, 10.14996253824105325591105570745, 10.93579246578082706164929704548, 11.325936545215755644882603616529, 12.28880276065504423981573408520, 12.54005260779133486399953237895, 13.17160872587143440059406854652, 14.398176647271825447711731865980, 14.66815297438822316983733234550, 15.38527329170753269711046013207, 16.09142235667119862567288046573, 16.86515378863978972131093594702, 17.63492714445242254250478671951, 17.93324254557450341355124041441

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