↑

Properties

Label 1-71-71.6-r0-0-0
Degree $1$
Conductor $71$
Sign $-0.623 - 0.782i$
Analytic cond. $0.329722$
Root an. cond. $0.329722$
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  + (−0.0448 − 0.998i)2-s + (0.753 − 0.657i)3-s + (−0.995 + 0.0896i)4-s + (0.309 − 0.951i)5-s + (−0.691 − 0.722i)6-s + (−0.963 + 0.266i)7-s + (0.134 + 0.990i)8-s + (0.134 − 0.990i)9-s + (−0.963 − 0.266i)10-s + (0.473 + 0.880i)11-s + (−0.691 + 0.722i)12-s + (0.473 − 0.880i)13-s + (0.309 + 0.951i)14-s + (−0.393 − 0.919i)15-s + (0.983 − 0.178i)16-s + (−0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (−0.0448 − 0.998i)2-s + (0.753 − 0.657i)3-s + (−0.995 + 0.0896i)4-s + (0.309 − 0.951i)5-s + (−0.691 − 0.722i)6-s + (−0.963 + 0.266i)7-s + (0.134 + 0.990i)8-s + (0.134 − 0.990i)9-s + (−0.963 − 0.266i)10-s + (0.473 + 0.880i)11-s + (−0.691 + 0.722i)12-s + (0.473 − 0.880i)13-s + (0.309 + 0.951i)14-s + (−0.393 − 0.919i)15-s + (0.983 − 0.178i)16-s + (−0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.623 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(71\)
Sign: $-0.623 - 0.782i$
Analytic conductor: \(0.329722\)
Root analytic conductor: \(0.329722\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 71,\ (0:\ ),\ -0.623 - 0.782i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4387298478 - 0.9103086137i\)
\(L(\frac12)\) \(\approx\) \(0.4387298478 - 0.9103086137i\)
\(L(1)\) \(\approx\) \(0.7708655054 - 0.7760046057i\)
\(L(1)\) \(\approx\) \(0.7708655054 - 0.7760046057i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 \)
good2 \( 1 + (-0.0448 - 0.998i)T \)
3 \( 1 + (0.753 - 0.657i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (-0.963 + 0.266i)T \)
11 \( 1 + (0.473 + 0.880i)T \)
13 \( 1 + (0.473 - 0.880i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.393 + 0.919i)T \)
23 \( 1 + (0.623 - 0.781i)T \)
29 \( 1 + (0.858 + 0.512i)T \)
31 \( 1 + (0.983 + 0.178i)T \)
37 \( 1 + (0.623 + 0.781i)T \)
41 \( 1 + (-0.900 + 0.433i)T \)
43 \( 1 + (0.936 - 0.351i)T \)
47 \( 1 + (0.753 + 0.657i)T \)
53 \( 1 + (-0.995 - 0.0896i)T \)
59 \( 1 + (-0.691 + 0.722i)T \)
61 \( 1 + (-0.963 - 0.266i)T \)
67 \( 1 + (-0.995 + 0.0896i)T \)
73 \( 1 + (-0.0448 - 0.998i)T \)
79 \( 1 + (0.134 + 0.990i)T \)
83 \( 1 + (-0.691 + 0.722i)T \)
89 \( 1 + (-0.995 - 0.0896i)T \)
97 \( 1 + (-0.900 - 0.433i)T \)
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

−32.311868397956763810045923220645, −31.338201798788270620297726313467, −30.20850974312186009874535452592, −28.69243488226761920415677294644, −27.1146019481123537193228514539, −26.473605552333832591749986768938, −25.74976949831260948719592815596, −24.79723657429111953194426560564, −23.27987926997586906667253786362, −22.17835568707354775267078639714, −21.47785150169720852628401730126, −19.49895344829509145579929530392, −18.856453002107834354096490616719, −17.27511218612994146275219876324, −16.06292321214883957452678638950, −15.28991873484917349190972221745, −13.88921714532953230457160674136, −13.54504427561819914709297440039, −10.9837673443878886813481247352, −9.6278629697208679496548697581, −8.83399119959582495108131759954, −7.1485838135212104194991971443, −6.17183099055675407627624456566, −4.26624471629431349580335325270, −3.014164056215514392948398287355, 1.36498050814056449195971163706, 2.81851501368607896615433222328, 4.31900479385879930019931395240, 6.24337823945671290634763054255, 8.25711453509361635155409848843, 9.12425292617385876433279124446, 10.21362879402753875487138513398, 12.33294143170552882372555926802, 12.71641483435411159004062385031, 13.74610663356964426751399540577, 15.25247075135542775664658437585, 17.075867679760703842365922467348, 18.16697207452872401940346579110, 19.379016802012120434023711221955, 20.14875695334514740082894072199, 20.94974821656626572132459162274, 22.41007842380812170325759965511, 23.50899760631191618978306260346, 25.01745523433226312541012656262, 25.69411980388208509464596455700, 27.15191817306370283269012665430, 28.45311001449174863740085265945, 29.09486983798162794739752611708, 30.26398313854723411507563073666, 31.145327596844939170265871984988

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