↑

Properties

Label 1-71e2-5041.1066-r0-0-0
Degree $1$
Conductor $5041$
Sign $0.947 + 0.319i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.598 + 0.801i)2-s + (−0.666 − 0.745i)3-s + (−0.283 − 0.958i)4-s + (−0.952 + 0.304i)5-s + (0.996 − 0.0883i)6-s + (0.699 − 0.714i)7-s + (0.937 + 0.346i)8-s + (−0.110 + 0.993i)9-s + (0.325 − 0.945i)10-s + 11-s + (−0.525 + 0.850i)12-s + (−0.598 − 0.801i)13-s + (0.154 + 0.988i)14-s + (0.862 + 0.506i)15-s + (−0.839 + 0.544i)16-s + (0.562 + 0.826i)17-s + ⋯
L(s)  = 1  + (−0.598 + 0.801i)2-s + (−0.666 − 0.745i)3-s + (−0.283 − 0.958i)4-s + (−0.952 + 0.304i)5-s + (0.996 − 0.0883i)6-s + (0.699 − 0.714i)7-s + (0.937 + 0.346i)8-s + (−0.110 + 0.993i)9-s + (0.325 − 0.945i)10-s + 11-s + (−0.525 + 0.850i)12-s + (−0.598 − 0.801i)13-s + (0.154 + 0.988i)14-s + (0.862 + 0.506i)15-s + (−0.839 + 0.544i)16-s + (0.562 + 0.826i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5041 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5041\)    =    \(71^{2}\)
Sign: $0.947 + 0.319i$
Analytic conductor: \(23.4103\)
Root analytic conductor: \(23.4103\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5041} (1066, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5041,\ (0:\ ),\ 0.947 + 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9613170705 + 0.1577590425i\)
\(L(\frac12)\) \(\approx\) \(0.9613170705 + 0.1577590425i\)
\(L(1)\) \(\approx\) \(0.6617661333 + 0.06388832661i\)
\(L(1)\) \(\approx\) \(0.6617661333 + 0.06388832661i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.598 + 0.801i)T \)
3 \( 1 + (-0.666 - 0.745i)T \)
5 \( 1 + (-0.952 + 0.304i)T \)
7 \( 1 + (0.699 - 0.714i)T \)
11 \( 1 + T \)
13 \( 1 + (-0.598 - 0.801i)T \)
17 \( 1 + (0.562 + 0.826i)T \)
19 \( 1 + (0.240 + 0.970i)T \)
23 \( 1 + (0.937 + 0.346i)T \)
29 \( 1 + (-0.110 - 0.993i)T \)
31 \( 1 + (-0.448 + 0.894i)T \)
37 \( 1 + (0.937 + 0.346i)T \)
41 \( 1 + (0.633 - 0.773i)T \)
43 \( 1 + (0.759 - 0.650i)T \)
47 \( 1 + (-0.197 - 0.980i)T \)
53 \( 1 + (-0.448 + 0.894i)T \)
59 \( 1 + (-0.787 + 0.616i)T \)
61 \( 1 + (0.903 + 0.428i)T \)
67 \( 1 + (0.699 - 0.714i)T \)
73 \( 1 + (0.996 + 0.0883i)T \)
79 \( 1 + (0.984 - 0.176i)T \)
83 \( 1 + (-0.197 + 0.980i)T \)
89 \( 1 + (0.862 - 0.506i)T \)
97 \( 1 + (-0.197 + 0.980i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.961944305974171162283796979192, −17.393103706984875599239676228220, −16.595728931359823209024174518143, −16.34085023195368890396923503400, −15.517444863154507518159757458977, −14.654413594203941974487010874352, −14.27889866757435134707671020185, −12.79186273619053309189393523343, −12.444432724516535943985648846213, −11.54275963192791632630639511912, −11.386600387955144754933723281, −10.98995000418670924264722297516, −9.66258760991771910538538689432, −9.26659588790007746883714998197, −8.883097134312565130370654178623, −7.88332750278879611640751614658, −7.2109442775926130042314565566, −6.41611519479763118816389064051, −5.10055147520507473219980060755, −4.73800358810237052296666531046, −4.10167518818739379570576236055, −3.26101861915913068300190808945, −2.5072471631052325020520153198, −1.31004115050212602957658855470, −0.62012600553511458933780824160, 0.74501987793586283100000883720, 1.19065704181676101742609552134, 2.17615029974347201415912971742, 3.5656388810725559925748222504, 4.2923267460455823826976711971, 5.13421907645309909400777515554, 5.78855274105544228530656127356, 6.62256455111801510724717742555, 7.21435641960270585754319869740, 7.84962387347783039306235577432, 8.05871506217857808802413635310, 9.07413065079937818571139721637, 10.13643115623034263530661395827, 10.67998890344256449923975771881, 11.22048296618407297926883447053, 12.014812868484607384883294518173, 12.57856090611150396555227485350, 13.57988342526543547321329398788, 14.29021180864237425060455131386, 14.78030534656386921132622917272, 15.403312366482588548718512030781, 16.35166774743485314814051939874, 16.93716885900142784742136227346, 17.262718999721647941555172850642, 17.93583488240503335530711481256

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