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Properties

Label 2-891-99.2-c1-0-37
Degree $2$
Conductor $891$
Sign $0.956 + 0.290i$
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  + (1.57 + 1.74i)2-s + (−0.368 + 3.50i)4-s + (−1.67 − 1.50i)5-s + (−1.64 − 3.70i)7-s + (−2.90 + 2.10i)8-s − 5.28i·10-s + (1.64 − 2.88i)11-s + (−0.967 − 4.55i)13-s + (3.87 − 8.69i)14-s + (−1.35 − 0.287i)16-s + (−0.0235 − 0.0725i)17-s + (1.40 + 1.93i)19-s + (5.89 − 5.30i)20-s + (7.61 − 1.66i)22-s + (−2.79 + 1.61i)23-s + ⋯
L(s)  = 1  + (1.11 + 1.23i)2-s + (−0.184 + 1.75i)4-s + (−0.747 − 0.672i)5-s + (−0.622 − 1.39i)7-s + (−1.02 + 0.745i)8-s − 1.67i·10-s + (0.494 − 0.869i)11-s + (−0.268 − 1.26i)13-s + (1.03 − 2.32i)14-s + (−0.337 − 0.0718i)16-s + (−0.00571 − 0.0175i)17-s + (0.323 + 0.444i)19-s + (1.31 − 1.18i)20-s + (1.62 − 0.355i)22-s + (−0.582 + 0.336i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.956 + 0.290i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.956 + 0.290i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93303 - 0.287398i\)
\(L(\frac12)\) \(\approx\) \(1.93303 - 0.287398i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.64 + 2.88i)T \)
good2 \( 1 + (-1.57 - 1.74i)T + (-0.209 + 1.98i)T^{2} \)
5 \( 1 + (1.67 + 1.50i)T + (0.522 + 4.97i)T^{2} \)
7 \( 1 + (1.64 + 3.70i)T + (-4.68 + 5.20i)T^{2} \)
13 \( 1 + (0.967 + 4.55i)T + (-11.8 + 5.28i)T^{2} \)
17 \( 1 + (0.0235 + 0.0725i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.40 - 1.93i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.79 - 1.61i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.68 - 0.748i)T + (19.4 - 21.5i)T^{2} \)
31 \( 1 + (1.63 - 0.348i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (5.87 + 4.27i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-7.71 - 3.43i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (3.70 + 2.14i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.18 + 0.649i)T + (45.9 - 9.77i)T^{2} \)
53 \( 1 + (-1.16 - 0.379i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-0.577 - 0.0606i)T + (57.7 + 12.2i)T^{2} \)
61 \( 1 + (-0.786 + 3.69i)T + (-55.7 - 24.8i)T^{2} \)
67 \( 1 + (6.49 + 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.06 + 0.346i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.82 + 10.7i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.490 - 0.441i)T + (8.25 - 78.5i)T^{2} \)
83 \( 1 + (-9.98 - 2.12i)T + (75.8 + 33.7i)T^{2} \)
89 \( 1 - 6.58iT - 89T^{2} \)
97 \( 1 + (-11.0 - 12.2i)T + (-10.1 + 96.4i)T^{2} \)
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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

−10.12886926104920543481341457329, −8.918602979834886238924254926067, −7.81518381553811039145630744666, −7.62018813734301079536684464358, −6.54522899168650180570144380647, −5.73660981249170533228620154570, −4.81377255906271261005062650895, −3.78506907316786452767410753227, −3.46492911099058457990743142091, −0.65206887222915574603637864585, 1.93767670400486316735746968917, 2.72511259380338390139389188933, 3.72231126970513998940923713057, 4.50390925835696018447923998329, 5.55551029286201597436879135693, 6.52275411374914694720938227935, 7.38727834523949803088507556083, 8.859366984910308454954779026969, 9.546423055012654974112087460681, 10.36083734462835329624760994309

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