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Properties

Label 2-891-99.70-c1-0-28
Degree $2$
Conductor $891$
Sign $0.988 - 0.153i$
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  + (−2.13 + 0.952i)2-s + (2.32 − 2.58i)4-s + (3.27 + 1.45i)5-s + (−1.12 − 0.239i)7-s + (−1.07 + 3.29i)8-s − 8.39·10-s + (3.31 + 0.0296i)11-s + (0.662 − 6.30i)13-s + (2.63 − 0.560i)14-s + (−0.120 − 1.14i)16-s + (3.16 − 2.29i)17-s + (0.457 − 1.40i)19-s + (11.4 − 5.07i)20-s + (−7.12 + 3.09i)22-s + (0.623 + 1.08i)23-s + ⋯
L(s)  = 1  + (−1.51 + 0.673i)2-s + (1.16 − 1.29i)4-s + (1.46 + 0.652i)5-s + (−0.425 − 0.0904i)7-s + (−0.378 + 1.16i)8-s − 2.65·10-s + (0.999 + 0.00892i)11-s + (0.183 − 1.74i)13-s + (0.704 − 0.149i)14-s + (−0.0300 − 0.286i)16-s + (0.767 − 0.557i)17-s + (0.104 − 0.322i)19-s + (2.55 − 1.13i)20-s + (−1.51 + 0.659i)22-s + (0.130 + 0.225i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.988 - 0.153i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.988 - 0.153i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.976884 + 0.0756026i\)
\(L(\frac12)\) \(\approx\) \(0.976884 + 0.0756026i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 - 0.0296i)T \)
good2 \( 1 + (2.13 - 0.952i)T + (1.33 - 1.48i)T^{2} \)
5 \( 1 + (-3.27 - 1.45i)T + (3.34 + 3.71i)T^{2} \)
7 \( 1 + (1.12 + 0.239i)T + (6.39 + 2.84i)T^{2} \)
13 \( 1 + (-0.662 + 6.30i)T + (-12.7 - 2.70i)T^{2} \)
17 \( 1 + (-3.16 + 2.29i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.457 + 1.40i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.623 - 1.08i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.51 + 1.17i)T + (26.4 + 11.7i)T^{2} \)
31 \( 1 + (-0.336 + 3.19i)T + (-30.3 - 6.44i)T^{2} \)
37 \( 1 + (3.09 + 9.53i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.89 + 0.403i)T + (37.4 - 16.6i)T^{2} \)
43 \( 1 + (-4.59 + 7.95i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.166 - 0.184i)T + (-4.91 + 46.7i)T^{2} \)
53 \( 1 + (4.71 + 3.42i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (4.42 - 4.90i)T + (-6.16 - 58.6i)T^{2} \)
61 \( 1 + (-0.344 - 3.27i)T + (-59.6 + 12.6i)T^{2} \)
67 \( 1 + (-7.03 - 12.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.25 - 3.09i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.683 - 2.10i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (2.71 - 1.20i)T + (52.8 - 58.7i)T^{2} \)
83 \( 1 + (-1.22 - 11.6i)T + (-81.1 + 17.2i)T^{2} \)
89 \( 1 - 8.93T + 89T^{2} \)
97 \( 1 + (-9.03 + 4.02i)T + (64.9 - 72.0i)T^{2} \)
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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

−9.831890321977447031823588875888, −9.465949348322008605921388904971, −8.628302040966269719813246352743, −7.49451448187821407326293352497, −6.97054068853105800199487260344, −5.89483257168413977832505618482, −5.62371835158137716344224411994, −3.46721286837278599766549673251, −2.20189491442193836450072819966, −0.863915680443460445790353336870, 1.40228471628589038027917064798, 1.83212793061219986161811508386, 3.29963122417792522038578205406, 4.74839172493327285613862591062, 6.11715707286057410786254279512, 6.67192279934838921669981369677, 7.928960204148635100725329604836, 8.964672539355700306981988343494, 9.303920893331682761183601200751, 9.777339145946142034092898043242

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