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Properties

Label 1-71e2-5041.19-r0-0-0
Degree $1$
Conductor $5041$
Sign $0.999 + 0.00124i$
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  + (−0.553 + 0.833i)2-s + (−0.998 + 0.0492i)3-s + (−0.388 − 0.921i)4-s + (−0.162 − 0.986i)5-s + (0.511 − 0.859i)6-s + (0.872 − 0.488i)7-s + (0.982 + 0.186i)8-s + (0.995 − 0.0984i)9-s + (0.912 + 0.409i)10-s + (0.858 + 0.512i)11-s + (0.433 + 0.901i)12-s + (0.935 − 0.353i)13-s + (−0.0751 + 0.997i)14-s + (0.211 + 0.977i)15-s + (−0.698 + 0.715i)16-s + (−0.979 + 0.202i)17-s + ⋯
L(s)  = 1  + (−0.553 + 0.833i)2-s + (−0.998 + 0.0492i)3-s + (−0.388 − 0.921i)4-s + (−0.162 − 0.986i)5-s + (0.511 − 0.859i)6-s + (0.872 − 0.488i)7-s + (0.982 + 0.186i)8-s + (0.995 − 0.0984i)9-s + (0.912 + 0.409i)10-s + (0.858 + 0.512i)11-s + (0.433 + 0.901i)12-s + (0.935 − 0.353i)13-s + (−0.0751 + 0.997i)14-s + (0.211 + 0.977i)15-s + (−0.698 + 0.715i)16-s + (−0.979 + 0.202i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5041\)    =    \(71^{2}\)
Sign: $0.999 + 0.00124i$
Analytic conductor: \(23.4103\)
Root analytic conductor: \(23.4103\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5041} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5041,\ (0:\ ),\ 0.999 + 0.00124i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9431494997 + 0.0005877786001i\)
\(L(\frac12)\) \(\approx\) \(0.9431494997 + 0.0005877786001i\)
\(L(1)\) \(\approx\) \(0.6661199185 + 0.07787843266i\)
\(L(1)\) \(\approx\) \(0.6661199185 + 0.07787843266i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.553 + 0.833i)T \)
3 \( 1 + (-0.998 + 0.0492i)T \)
5 \( 1 + (-0.162 - 0.986i)T \)
7 \( 1 + (0.872 - 0.488i)T \)
11 \( 1 + (0.858 + 0.512i)T \)
13 \( 1 + (0.935 - 0.353i)T \)
17 \( 1 + (-0.979 + 0.202i)T \)
19 \( 1 + (0.537 - 0.843i)T \)
23 \( 1 + (-0.557 + 0.830i)T \)
29 \( 1 + (-0.939 + 0.343i)T \)
31 \( 1 + (-0.984 - 0.177i)T \)
37 \( 1 + (-0.996 + 0.0820i)T \)
41 \( 1 + (0.892 - 0.450i)T \)
43 \( 1 + (0.635 + 0.772i)T \)
47 \( 1 + (-0.991 + 0.127i)T \)
53 \( 1 + (-0.132 + 0.991i)T \)
59 \( 1 + (0.830 + 0.557i)T \)
61 \( 1 + (0.504 + 0.863i)T \)
67 \( 1 + (-0.912 + 0.408i)T \)
73 \( 1 + (0.589 - 0.807i)T \)
79 \( 1 + (0.554 + 0.832i)T \)
83 \( 1 + (-0.0827 + 0.996i)T \)
89 \( 1 + (-0.745 + 0.666i)T \)
97 \( 1 + (0.652 - 0.757i)T \)
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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

−17.96912837726182142937537741840, −17.77611259889839077120961545058, −16.86821142290318305197426571604, −16.187837908579373538580713155143, −15.64356483943984450124216290762, −14.51840242889648439865341823614, −14.07295827737679583679817329237, −13.17649614521199834054102234833, −12.37650492928640743455322348885, −11.56522512116913557214609662973, −11.399598743945041007259838343022, −10.859723363024656095015740673361, −10.18516207169214659209518956918, −9.31786979979969633897759764051, −8.63066185862920205373748789465, −7.85675585793463436633099444106, −7.10258099561593180819495061728, −6.37851410532106271180300824422, −5.70974165996591194295565227529, −4.69652990304769074288360389486, −3.886489094321827204901975147788, −3.44314354008283830489878848210, −2.01842352087325073070882699289, −1.82306077849722853789289237062, −0.64047594161708824554257070744, 0.58826945308085460668768715821, 1.39635661262018760098913073544, 1.78429476326278417799687973011, 3.961410542665694489545348758212, 4.21823253692793341605293484200, 5.11307491902798799152196480929, 5.55958220620966547277316011533, 6.338857073121131249272228812313, 7.21991367538288612372918051774, 7.594374323725814074085722143795, 8.564805407064863992671505681983, 9.156598431819664601018788318602, 9.73342541268880299857474436266, 10.771020414945715004607631035909, 11.181280376543310010851201696657, 11.793463272353844366393863809369, 12.809255738658954850365693132777, 13.37082971507944725257752663087, 14.05645831647948881091161948972, 15.06120013783044162754052360415, 15.53804280938983917342816605602, 16.22709114555392007045799593211, 16.72294349408445026597377472634, 17.44554437667290444154468409651, 17.824222105614461286606479274831

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