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Properties

Label 1-71e2-5041.54-r0-0-0
Degree $1$
Conductor $5041$
Sign $-0.921 + 0.389i$
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  + (−0.914 − 0.404i)2-s + (0.931 + 0.363i)3-s + (0.673 + 0.739i)4-s + (−0.765 + 0.643i)5-s + (−0.705 − 0.708i)6-s + (0.136 + 0.990i)7-s + (−0.317 − 0.948i)8-s + (0.736 + 0.676i)9-s + (0.960 − 0.279i)10-s + (0.309 + 0.951i)11-s + (0.359 + 0.933i)12-s + (−0.914 + 0.404i)13-s + (0.275 − 0.961i)14-s + (−0.946 + 0.321i)15-s + (−0.0927 + 0.995i)16-s + (−0.432 + 0.901i)17-s + ⋯
L(s)  = 1  + (−0.914 − 0.404i)2-s + (0.931 + 0.363i)3-s + (0.673 + 0.739i)4-s + (−0.765 + 0.643i)5-s + (−0.705 − 0.708i)6-s + (0.136 + 0.990i)7-s + (−0.317 − 0.948i)8-s + (0.736 + 0.676i)9-s + (0.960 − 0.279i)10-s + (0.309 + 0.951i)11-s + (0.359 + 0.933i)12-s + (−0.914 + 0.404i)13-s + (0.275 − 0.961i)14-s + (−0.946 + 0.321i)15-s + (−0.0927 + 0.995i)16-s + (−0.432 + 0.901i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5041\)    =    \(71^{2}\)
Sign: $-0.921 + 0.389i$
Analytic conductor: \(23.4103\)
Root analytic conductor: \(23.4103\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5041} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5041,\ (0:\ ),\ -0.921 + 0.389i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2661962310 + 1.314112665i\)
\(L(\frac12)\) \(\approx\) \(0.2661962310 + 1.314112665i\)
\(L(1)\) \(\approx\) \(0.7930781600 + 0.4161828855i\)
\(L(1)\) \(\approx\) \(0.7930781600 + 0.4161828855i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.914 - 0.404i)T \)
3 \( 1 + (0.931 + 0.363i)T \)
5 \( 1 + (-0.765 + 0.643i)T \)
7 \( 1 + (0.136 + 0.990i)T \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.914 + 0.404i)T \)
17 \( 1 + (-0.432 + 0.901i)T \)
19 \( 1 + (0.424 - 0.905i)T \)
23 \( 1 + (0.814 + 0.580i)T \)
29 \( 1 + (0.736 - 0.676i)T \)
31 \( 1 + (0.994 + 0.105i)T \)
37 \( 1 + (0.814 + 0.580i)T \)
41 \( 1 + (-0.0221 + 0.999i)T \)
43 \( 1 + (-0.819 + 0.573i)T \)
47 \( 1 + (-0.350 + 0.936i)T \)
53 \( 1 + (-0.866 + 0.498i)T \)
59 \( 1 + (-0.249 + 0.968i)T \)
61 \( 1 + (-0.463 - 0.885i)T \)
67 \( 1 + (-0.899 + 0.436i)T \)
73 \( 1 + (0.987 - 0.158i)T \)
79 \( 1 + (0.949 + 0.313i)T \)
83 \( 1 + (0.834 + 0.551i)T \)
89 \( 1 + (0.955 - 0.296i)T \)
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

−17.77259698670031301337961411908, −16.98253091712009886815929370620, −16.45548796733102866743859607243, −15.90750367944115570517580482357, −15.13363481417346322171118492321, −14.48226405142886063732923365246, −13.915560563720175921157507038666, −13.23163843489200092068805000751, −12.25168238066532146316749650497, −11.7432398049536172172707404854, −10.83235729961335473648033834990, −10.17818272937712332848781612862, −9.37043573400521377186053152030, −8.82722445088654554864590059941, −8.081164178963725964741755079074, −7.742237623911786747026513136904, −7.01076159355874425924949588217, −6.478373492764387478575157395912, −5.21140636809952800683109975212, −4.59578967497067015940247358240, −3.54536341412492058520052891840, −2.95029039062587538001698159443, −1.89791278783495680055832455402, −0.91432009800177065747736836274, −0.51891306261517598982392665801, 1.30854995615733162538794765542, 2.1699833921355702879585252579, 2.79630128453403898477466294859, 3.22051776638937366967618389848, 4.448074762123836281626946138869, 4.68444461449026394727747220834, 6.38216554841382600817741969133, 6.85946889546508027854436086384, 7.794429557246248173011289504574, 8.04726990757101112763801753149, 8.95817188613515940060818405289, 9.526251933330790763748342714782, 9.97757533010605712084116508900, 10.90559180647672814311231605936, 11.518699065746232101512174717365, 12.14331882045754797261456999367, 12.77403823124369957252948781214, 13.66191467412671600037283027195, 14.760409841336184837392301974905, 15.15065426337050268400418220640, 15.45477508777245496985980878984, 16.23844872632430216409803546002, 17.16383959825769676583194165754, 17.81169142900591895961413724897, 18.44363555169255176757345259748

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