↑

Properties

Label 1-671-671.145-r0-0-0
Degree $1$
Conductor $671$
Sign $-0.735 + 0.677i$
Analytic cond. $3.11611$
Root an. cond. $3.11611$
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  + (0.951 + 0.309i)2-s − 3-s + (0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.951 − 0.309i)6-s + i·7-s + (0.587 + 0.809i)8-s + 9-s + (−0.587 + 0.809i)10-s + (−0.809 − 0.587i)12-s + (0.809 + 0.587i)13-s + (−0.309 + 0.951i)14-s + (0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.951 − 0.309i)17-s + (0.951 + 0.309i)18-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s − 3-s + (0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.951 − 0.309i)6-s + i·7-s + (0.587 + 0.809i)8-s + 9-s + (−0.587 + 0.809i)10-s + (−0.809 − 0.587i)12-s + (0.809 + 0.587i)13-s + (−0.309 + 0.951i)14-s + (0.309 − 0.951i)15-s + (0.309 + 0.951i)16-s + (−0.951 − 0.309i)17-s + (0.951 + 0.309i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $-0.735 + 0.677i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ -0.735 + 0.677i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6402782453 + 1.639552412i\)
\(L(\frac12)\) \(\approx\) \(0.6402782453 + 1.639552412i\)
\(L(1)\) \(\approx\) \(1.114822177 + 0.8030088706i\)
\(L(1)\) \(\approx\) \(1.114822177 + 0.8030088706i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.951 + 0.309i)T \)
3 \( 1 - T \)
5 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + iT \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + T \)
23 \( 1 + (0.587 + 0.809i)T \)
29 \( 1 + (0.951 - 0.309i)T \)
31 \( 1 + (-0.587 - 0.809i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.951 - 0.309i)T \)
59 \( 1 + (-0.587 - 0.809i)T \)
67 \( 1 + (-0.587 - 0.809i)T \)
71 \( 1 + iT \)
73 \( 1 + (-0.309 + 0.951i)T \)
79 \( 1 + (-0.951 + 0.309i)T \)
83 \( 1 + (0.809 + 0.587i)T \)
89 \( 1 + (-0.951 + 0.309i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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

−22.64353202850830580735311668164, −21.72871222097183708221893294607, −20.84856680623233525213508027391, −20.212001754875527897803134175051, −19.55500523274984668609060069817, −18.27666772706342661596062699170, −17.39188319655167617932753346993, −16.39060780289404892350849199271, −16.03778592787369668322031955817, −15.11627581410489623620899824584, −13.78075265629364269161331498385, −13.166867368385384700764538811169, −12.48258168935914530669137534041, −11.656452852281652915445738847101, −10.78701094636220012085162671087, −10.28795058931997116075848894554, −8.92594743509513140249846707142, −7.57164733659615822632287502574, −6.75140606091937081504090353301, −5.783871416498688267248967793221, −4.88478478905191165569867398093, −4.28072595960692472225357115677, −3.327677824233713786120690207502, −1.53410442139455011016776311850, −0.75228442756164330918159580593, 1.763200633014240591393771082587, 2.91496450009197822056418467206, 3.9111729877943390474519872272, 4.96603302650219401031777048004, 5.80810965454435077504532831211, 6.58818348623510495572925611407, 7.18834805111537528630438139803, 8.39015006120328024748787630369, 9.73326685820279885332085956910, 10.94703683273634670919396666085, 11.57345892859008521039930093039, 11.931829119407992354898723007199, 13.17133701496018086188098054443, 13.84183340640732652259526223349, 15.07026571381402459895111140476, 15.565792090897665003505135615287, 16.17384309320410107842723150213, 17.26232357764545133267425407284, 18.19541442331157535454508859801, 18.72951468892879310942164064168, 19.89837405752542411926546706523, 21.09873775695706453882676793474, 21.79936450454408399293562030481, 22.29489536317011232817618306656, 23.003266371100136104870483114164

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