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Properties

Label 2-891-99.65-c1-0-1
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  + (1 + 1.73i)4-s + (−2.87 + 1.65i)5-s + (−2.87 − 1.65i)11-s + (−1.99 + 3.46i)16-s + (−5.74 − 3.31i)20-s + (−2.87 + 1.65i)23-s + (3 − 5.19i)25-s + (−2.5 − 4.33i)31-s − 7·37-s − 6.63i·44-s + (−5.74 − 3.31i)47-s + (−3.5 − 6.06i)49-s − 13.2i·53-s + 11·55-s + (−2.87 + 1.65i)59-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)4-s + (−1.28 + 0.741i)5-s + (−0.866 − 0.500i)11-s + (−0.499 + 0.866i)16-s + (−1.28 − 0.741i)20-s + (−0.598 + 0.345i)23-s + (0.600 − 1.03i)25-s + (−0.449 − 0.777i)31-s − 1.15·37-s − 1.00i·44-s + (−0.837 − 0.483i)47-s + (−0.5 − 0.866i)49-s − 1.82i·53-s + 1.48·55-s + (−0.373 + 0.215i)59-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: $\chi_{891} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ -0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0586169 - 0.332433i\)
\(L(\frac12)\) \(\approx\) \(0.0586169 - 0.332433i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.87 + 1.65i)T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
5 \( 1 + (2.87 - 1.65i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
13 \( 1 + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (2.87 - 1.65i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 7T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.74 + 3.31i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 13.2iT - 53T^{2} \)
59 \( 1 + (2.87 - 1.65i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 16.5iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 16.5iT - 89T^{2} \)
97 \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.84753272482847562195380418880, −9.911838511652195635510271210342, −8.467380098947631633916828948841, −8.062378431467771924990753052382, −7.27280629085584846360061588268, −6.62093092724160990050062956807, −5.33138232778613268813350963824, −3.93550526785674395803136813177, −3.39168799879192334715402104609, −2.34215126330491853152186341114, 0.15201786190032731353748668638, 1.69088087249798767184320277943, 3.13220723415575941474233084906, 4.47051339389215530939877516809, 5.07858078151097757430794097752, 6.15818472050728506820835450243, 7.24901565549000233047394663948, 7.86959163882517146026709359254, 8.779739315869227476974687921446, 9.716689691559407121104597561179

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