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Properties

Label 1-71e2-5041.1847-r0-0-0
Degree $1$
Conductor $5041$
Sign $0.937 - 0.348i$
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  + (0.937 − 0.346i)2-s + (0.814 + 0.580i)3-s + (0.759 − 0.650i)4-s + (−0.598 − 0.801i)5-s + (0.964 + 0.262i)6-s + (−0.730 + 0.683i)7-s + (0.487 − 0.873i)8-s + (0.325 + 0.945i)9-s + (−0.839 − 0.544i)10-s + 11-s + (0.996 − 0.0883i)12-s + (0.937 + 0.346i)13-s + (−0.448 + 0.894i)14-s + (−0.0221 − 0.999i)15-s + (0.154 − 0.988i)16-s + (−0.975 − 0.219i)17-s + ⋯
L(s)  = 1  + (0.937 − 0.346i)2-s + (0.814 + 0.580i)3-s + (0.759 − 0.650i)4-s + (−0.598 − 0.801i)5-s + (0.964 + 0.262i)6-s + (−0.730 + 0.683i)7-s + (0.487 − 0.873i)8-s + (0.325 + 0.945i)9-s + (−0.839 − 0.544i)10-s + 11-s + (0.996 − 0.0883i)12-s + (0.937 + 0.346i)13-s + (−0.448 + 0.894i)14-s + (−0.0221 − 0.999i)15-s + (0.154 − 0.988i)16-s + (−0.975 − 0.219i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5041\)    =    \(71^{2}\)
Sign: $0.937 - 0.348i$
Analytic conductor: \(23.4103\)
Root analytic conductor: \(23.4103\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5041} (1847, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5041,\ (0:\ ),\ 0.937 - 0.348i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.214815112 - 0.7592960048i\)
\(L(\frac12)\) \(\approx\) \(4.214815112 - 0.7592960048i\)
\(L(1)\) \(\approx\) \(2.244773351 - 0.2685688713i\)
\(L(1)\) \(\approx\) \(2.244773351 - 0.2685688713i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (0.937 - 0.346i)T \)
3 \( 1 + (0.814 + 0.580i)T \)
5 \( 1 + (-0.598 - 0.801i)T \)
7 \( 1 + (-0.730 + 0.683i)T \)
11 \( 1 + T \)
13 \( 1 + (0.937 + 0.346i)T \)
17 \( 1 + (-0.975 - 0.219i)T \)
19 \( 1 + (-0.666 + 0.745i)T \)
23 \( 1 + (0.487 - 0.873i)T \)
29 \( 1 + (0.325 - 0.945i)T \)
31 \( 1 + (0.984 + 0.176i)T \)
37 \( 1 + (0.487 - 0.873i)T \)
41 \( 1 + (-0.883 + 0.467i)T \)
43 \( 1 + (-0.525 + 0.850i)T \)
47 \( 1 + (0.562 - 0.826i)T \)
53 \( 1 + (0.984 + 0.176i)T \)
59 \( 1 + (0.408 - 0.912i)T \)
61 \( 1 + (0.240 - 0.970i)T \)
67 \( 1 + (-0.730 + 0.683i)T \)
73 \( 1 + (0.964 - 0.262i)T \)
79 \( 1 + (0.862 + 0.506i)T \)
83 \( 1 + (0.562 + 0.826i)T \)
89 \( 1 + (-0.0221 + 0.999i)T \)
97 \( 1 + (0.562 + 0.826i)T \)
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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

−18.020257080997249661933606837292, −17.38187529397653701675937014822, −16.63939605904968534459563719840, −15.6606378658661295541587496984, −15.32777528234156040234767708818, −14.7949680474744722167208509115, −13.88063626040538097042964659398, −13.54290805021744894111622753487, −13.03512984651877641396274515560, −12.15988936596343347214372548425, −11.54752695748418771974046834637, −10.83324108492086170605592576685, −10.12476598881962524043898513651, −8.88315750954492614750807195473, −8.52799137883714132630965535136, −7.508171805378044954201781516519, −6.98921951041768189849769352978, −6.5341640494835932350560961767, −6.04224558457626652484293458354, −4.56672130011224609321728610365, −3.97120380824138386720634102607, −3.371961406169019501103299568570, −2.90826374876519464295921935321, −1.95015893407887431109625467628, −0.91416154376026067340600416627, 0.87047597909804937093734767941, 1.92708621020847322753740710561, 2.54885281386389215899078963030, 3.55879819072897650225068746544, 3.9464547547631849730827020565, 4.55426210723324063936935380449, 5.29712317294164240856985294965, 6.38346669672969141122331886838, 6.65584309217549973819385569301, 7.96359333019601095826795803327, 8.63284517550163632066202401041, 9.16166701898595309047467975348, 9.840193526923614759088162368654, 10.68106539306894594314065474036, 11.48163513186802787019176490599, 12.0027449371375697262458041317, 12.79873133632590834918404231681, 13.29408824935768672109547138518, 13.925379336091198122158685642538, 14.758757477895553109103701014211, 15.312632871602722575176301367151, 15.7668136114488794572435859506, 16.47582193791210140304340515804, 16.83006284079405533348355592307, 18.30748037329185194341875707627

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