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Properties

Label 2-891-33.2-c1-0-4
Degree $2$
Conductor $891$
Sign $0.963 - 0.269i$
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  + (0.764 − 2.35i)2-s + (−3.32 − 2.41i)4-s + (−1.94 + 0.631i)5-s + (−0.637 + 0.877i)7-s + (−4.22 + 3.07i)8-s + 5.05i·10-s + (−2.00 + 2.64i)11-s + (1.05 + 0.343i)13-s + (1.57 + 2.16i)14-s + (1.44 + 4.46i)16-s + (2.03 + 6.25i)17-s + (−4.19 − 5.76i)19-s + (7.99 + 2.59i)20-s + (4.68 + 6.72i)22-s + 5.43i·23-s + ⋯
L(s)  = 1  + (0.540 − 1.66i)2-s + (−1.66 − 1.20i)4-s + (−0.869 + 0.282i)5-s + (−0.240 + 0.331i)7-s + (−1.49 + 1.08i)8-s + 1.59i·10-s + (−0.603 + 0.797i)11-s + (0.293 + 0.0952i)13-s + (0.421 + 0.579i)14-s + (0.362 + 1.11i)16-s + (0.493 + 1.51i)17-s + (−0.961 − 1.32i)19-s + (1.78 + 0.581i)20-s + (0.999 + 1.43i)22-s + 1.13i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.269i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.963 - 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.963 - 0.269i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (728, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.963 - 0.269i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.681143 + 0.0933912i\)
\(L(\frac12)\) \(\approx\) \(0.681143 + 0.0933912i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.00 - 2.64i)T \)
good2 \( 1 + (-0.764 + 2.35i)T + (-1.61 - 1.17i)T^{2} \)
5 \( 1 + (1.94 - 0.631i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (0.637 - 0.877i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.05 - 0.343i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.03 - 6.25i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.19 + 5.76i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 - 5.43iT - 23T^{2} \)
29 \( 1 + (-5.99 - 4.35i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (0.366 - 1.12i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-0.780 - 0.567i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (0.893 - 0.649i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 5.11iT - 43T^{2} \)
47 \( 1 + (2.52 + 3.47i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (10.3 + 3.36i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.29 - 1.78i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.51 - 1.79i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 0.993T + 67T^{2} \)
71 \( 1 + (6.12 - 1.98i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.486 - 0.669i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-5.79 - 1.88i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.677 - 2.08i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 6.19iT - 89T^{2} \)
97 \( 1 + (2.94 - 9.05i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.55893932942808069590839630495, −9.627099719910998626955561670021, −8.755904049180750016052436365052, −7.78840187990977516336638848493, −6.63455845327415633994286283625, −5.37135046324368926062556923287, −4.44721105268635936119039753135, −3.63648299321143584825185967839, −2.76403428648195986741946084341, −1.59835108387150432425052127510, 0.27850843199678834804853195913, 3.09112653904930549887237697954, 4.15613308525767539463327611348, 4.82657215875392647233503925905, 5.92498707038919497167585832218, 6.53064311585485189523595426480, 7.66142784400637888310661275891, 8.047140685703178410729529726060, 8.726570452249805126348168087398, 9.898638419093624253743348091822

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