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Properties

Label 1-927-927.209-r0-0-0
Degree $1$
Conductor $927$
Sign $0.497 + 0.867i$
Analytic cond. $4.30496$
Root an. cond. $4.30496$
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  + (0.998 − 0.0615i)2-s + (0.992 − 0.122i)4-s + (0.213 + 0.976i)5-s + (0.332 + 0.943i)7-s + (0.982 − 0.183i)8-s + (0.273 + 0.961i)10-s + (−0.998 + 0.0615i)11-s + (0.650 − 0.759i)13-s + (0.389 + 0.920i)14-s + (0.969 − 0.243i)16-s + (−0.0922 + 0.995i)17-s + (−0.850 − 0.526i)19-s + (0.332 + 0.943i)20-s + (−0.992 + 0.122i)22-s + (0.998 + 0.0615i)23-s + ⋯
L(s)  = 1  + (0.998 − 0.0615i)2-s + (0.992 − 0.122i)4-s + (0.213 + 0.976i)5-s + (0.332 + 0.943i)7-s + (0.982 − 0.183i)8-s + (0.273 + 0.961i)10-s + (−0.998 + 0.0615i)11-s + (0.650 − 0.759i)13-s + (0.389 + 0.920i)14-s + (0.969 − 0.243i)16-s + (−0.0922 + 0.995i)17-s + (−0.850 − 0.526i)19-s + (0.332 + 0.943i)20-s + (−0.992 + 0.122i)22-s + (0.998 + 0.0615i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.497 + 0.867i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ 0.497 + 0.867i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.657058517 + 1.538485088i\)
\(L(\frac12)\) \(\approx\) \(2.657058517 + 1.538485088i\)
\(L(1)\) \(\approx\) \(1.997288442 + 0.5219801563i\)
\(L(1)\) \(\approx\) \(1.997288442 + 0.5219801563i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.998 - 0.0615i)T \)
5 \( 1 + (0.213 + 0.976i)T \)
7 \( 1 + (0.332 + 0.943i)T \)
11 \( 1 + (-0.998 + 0.0615i)T \)
13 \( 1 + (0.650 - 0.759i)T \)
17 \( 1 + (-0.0922 + 0.995i)T \)
19 \( 1 + (-0.850 - 0.526i)T \)
23 \( 1 + (0.998 + 0.0615i)T \)
29 \( 1 + (0.952 + 0.303i)T \)
31 \( 1 + (0.696 + 0.717i)T \)
37 \( 1 + (-0.932 - 0.361i)T \)
41 \( 1 + (0.952 - 0.303i)T \)
43 \( 1 + (0.779 + 0.626i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.850 - 0.526i)T \)
59 \( 1 + (-0.332 + 0.943i)T \)
61 \( 1 + (-0.908 + 0.417i)T \)
67 \( 1 + (-0.332 + 0.943i)T \)
71 \( 1 + (0.739 + 0.673i)T \)
73 \( 1 + (-0.739 - 0.673i)T \)
79 \( 1 + (-0.952 - 0.303i)T \)
83 \( 1 + (-0.332 - 0.943i)T \)
89 \( 1 + (-0.602 + 0.798i)T \)
97 \( 1 + (0.816 - 0.577i)T \)
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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

−21.42600868386005299845684212266, −20.873226869170780551774343569257, −20.67457492377558538592334480229, −19.618316183683919451784909874908, −18.72588151549224438169568371089, −17.427855607668701448283461060904, −16.85564087828930799455156899436, −16.03604010289755730811489484926, −15.50375413853327076657082389555, −14.18161588552971141385513880628, −13.75199426339397799196571639071, −13.02350826341764717493399098691, −12.30243969155148538465735927486, −11.2785880547656326840288767470, −10.65092537767619595694512765184, −9.60394723119934992179329614946, −8.39561102428377612668246506925, −7.6823304994492138494933415143, −6.67763293050423056567506097808, −5.787807175081614530126275944042, −4.65433846155487818092441334626, −4.46619588795531922051060248700, −3.17460373435358369619307042282, −2.03248811559974848448364584801, −0.98325577356975862311112019965, 1.63062407861924834077873190639, 2.682029632021701780041261027131, 3.11115116242826329585470517972, 4.4185196709886857470802591740, 5.408645280311317345650481891226, 6.05402717687438288743109275901, 6.874885683500993429038711645247, 7.90843264679125196164365073244, 8.76607136314455319750652259653, 10.35126933434982490170637368458, 10.67938280948398736664089791335, 11.49107158135419535080377169345, 12.59440449539654612979390916665, 13.077517324948324296440421496500, 14.06266018165363717689446943081, 14.89426714320963193295092002108, 15.39877672655443104936618198831, 15.968633331935977664827354278308, 17.42884465862111697747136152945, 17.99414276606186639564126970965, 19.04874353907865760835690167456, 19.527917241994476213428039649687, 20.897225508895930053634115378767, 21.30269770437551574959360604190, 21.877946747533886920710292500364

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