↑

Properties

Label 1-861-861.524-r0-0-0
Degree $1$
Conductor $861$
Sign $0.331 + 0.943i$
Analytic cond. $3.99846$
Root an. cond. $3.99846$
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  + 2-s + 4-s − 5-s + 8-s − 10-s + i·11-s + i·13-s + 16-s + i·17-s − i·19-s − 20-s + i·22-s − 23-s + 25-s + i·26-s + ⋯
L(s)  = 1  + 2-s + 4-s − 5-s + 8-s − 10-s + i·11-s + i·13-s + 16-s + i·17-s − i·19-s − 20-s + i·22-s − 23-s + 25-s + i·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(861\)    =    \(3 \cdot 7 \cdot 41\)
Sign: $0.331 + 0.943i$
Analytic conductor: \(3.99846\)
Root analytic conductor: \(3.99846\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{861} (524, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 861,\ (0:\ ),\ 0.331 + 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.767479367 + 1.253069035i\)
\(L(\frac12)\) \(\approx\) \(1.767479367 + 1.253069035i\)
\(L(1)\) \(\approx\) \(1.589603913 + 0.3530832894i\)
\(L(1)\) \(\approx\) \(1.589603913 + 0.3530832894i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
41 \( 1 \)
good2 \( 1 \)
5 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
17 \( 1 \)
19 \( 1 \)
23 \( 1 + T \)
29 \( 1 \)
31 \( 1 - T \)
37 \( 1 + iT \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 + T \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 - iT \)
73 \( 1 - T \)
79 \( 1 \)
83 \( 1 + iT \)
89 \( 1 - T \)
97 \( 1 \)
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

−22.13223098770157619715024404504, −21.134607012142341687646242570481, −20.33578837809524063920260658829, −19.80229882003119665892658982930, −18.89596665580660832032431817895, −18.06819765023634595154082144080, −16.56923791516984112990892732272, −16.27991323172266913309894053782, −15.37299043274541641892958509126, −14.700035047538261250303728397184, −13.81585442644982229225949198614, −13.03643841071074552112060305180, −12.118323946887964962822706310639, −11.5296419863353649428178204918, −10.77596173387016866979262019702, −9.79987262447290047202349062822, −8.229134817180741600796464165094, −7.85209515860951260070500157974, −6.7727013864339777859153446620, −5.786826812723731445502802972520, −5.039492171567203643807727328605, −3.84364978847679906119214957593, −3.3866891889732090274243651703, −2.27207211611163152896332612142, −0.69189950075183436600003873732, 1.52309403706327487797313311379, 2.53714251435897216724625098013, 3.764143959160724146519671177452, 4.303218107160619619715549924880, 5.14503527141934851451972699948, 6.40284330306643662542458287277, 7.10131874543822702866046801746, 7.859992898865770273684043199831, 8.96484689154316333199253044026, 10.1743948739919451637917823574, 11.12023299841340358754539448344, 11.74771421700038666166194799771, 12.55024364846094037446474634327, 13.15848569817929130285177149850, 14.416161397044960704086058409338, 14.816938527511606914166318705727, 15.71875690829247075560815685366, 16.31599909885051674779710705201, 17.224665158360562079059097783733, 18.35799038985356725036151239434, 19.39342972389431100406751991456, 19.95769758982572868291214449967, 20.55341613031707766751288542852, 21.74791515121004186155477421367, 22.088377403407637040337317756

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