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Properties

Label 2-891-9.7-c1-0-10
Degree $2$
Conductor $891$
Sign $0.939 - 0.342i$
Analytic cond. $7.11467$
Root an. cond. $2.66733$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.500 + 0.866i)4-s + (−1 − 1.73i)5-s + (−2 + 3.46i)7-s + 3·8-s − 1.99·10-s + (0.5 − 0.866i)11-s + (1 + 1.73i)13-s + (1.99 + 3.46i)14-s + (0.500 − 0.866i)16-s + 2·17-s + (0.999 − 1.73i)20-s + (−0.499 − 0.866i)22-s + (4 + 6.92i)23-s + (0.500 − 0.866i)25-s + 1.99·26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.250 + 0.433i)4-s + (−0.447 − 0.774i)5-s + (−0.755 + 1.30i)7-s + 1.06·8-s − 0.632·10-s + (0.150 − 0.261i)11-s + (0.277 + 0.480i)13-s + (0.534 + 0.925i)14-s + (0.125 − 0.216i)16-s + 0.485·17-s + (0.223 − 0.387i)20-s + (−0.106 − 0.184i)22-s + (0.834 + 1.44i)23-s + (0.100 − 0.173i)25-s + 0.392·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.80195 + 0.317733i\)
\(L(\frac12)\) \(\approx\) \(1.80195 + 0.317733i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (-4 - 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (1 + 1.73i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 14T + 73T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.26205031788557261912072099837, −9.132410686771526839332357109509, −8.727124020242786316614315104357, −7.71101318737073414285787821404, −6.73601811624836872724165904567, −5.64035159893455773232008609140, −4.74135430231099498418151920390, −3.57403440866074713131758552083, −2.87560877548079138827309625847, −1.49687613688693668414753015426, 0.856145449798869226071935477126, 2.73600333415952620808292653054, 3.87745152374594180562195107457, 4.68609387465894310633434542259, 6.07105715749106229903630370057, 6.56887328227304386772173689193, 7.42498722599783234180767426532, 7.88573859324328509506318435503, 9.466938437780251318043288735939, 10.22049985721243641980253240623

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