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Properties

Label 1-861-861.419-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

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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: $\chi_{861} (419, \cdot )$
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 \( 1 \)
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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

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

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