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Properties

Label 2-891-99.97-c1-0-41
Degree $2$
Conductor $891$
Sign $0.577 + 0.816i$
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  + (0.413 + 0.459i)2-s + (0.169 − 1.60i)4-s + (1.75 − 1.94i)5-s + (2.74 − 1.22i)7-s + (1.80 − 1.31i)8-s + 1.61·10-s + (3.31 − 0.0378i)11-s + (−1.72 + 0.366i)13-s + (1.69 + 0.754i)14-s + (−1.81 − 0.385i)16-s + (−0.5 − 1.53i)17-s + (−4.73 + 3.44i)19-s + (−2.83 − 3.14i)20-s + (1.38 + 1.50i)22-s + (1.73 + 3.00i)23-s + ⋯
L(s)  = 1  + (0.292 + 0.324i)2-s + (0.0845 − 0.804i)4-s + (0.783 − 0.870i)5-s + (1.03 − 0.461i)7-s + (0.639 − 0.464i)8-s + 0.511·10-s + (0.999 − 0.0114i)11-s + (−0.478 + 0.101i)13-s + (0.452 + 0.201i)14-s + (−0.453 − 0.0963i)16-s + (−0.121 − 0.373i)17-s + (−1.08 + 0.789i)19-s + (−0.633 − 0.703i)20-s + (0.296 + 0.321i)22-s + (0.361 + 0.626i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.577 + 0.816i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (460, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.577 + 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16352 - 1.11965i\)
\(L(\frac12)\) \(\approx\) \(2.16352 - 1.11965i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.31 + 0.0378i)T \)
good2 \( 1 + (-0.413 - 0.459i)T + (-0.209 + 1.98i)T^{2} \)
5 \( 1 + (-1.75 + 1.94i)T + (-0.522 - 4.97i)T^{2} \)
7 \( 1 + (-2.74 + 1.22i)T + (4.68 - 5.20i)T^{2} \)
13 \( 1 + (1.72 - 0.366i)T + (11.8 - 5.28i)T^{2} \)
17 \( 1 + (0.5 + 1.53i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (4.73 - 3.44i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.73 - 3.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.08 - 1.81i)T + (19.4 - 21.5i)T^{2} \)
31 \( 1 + (2.79 - 0.593i)T + (28.3 - 12.6i)T^{2} \)
37 \( 1 + (0.190 + 0.138i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-10.9 - 4.85i)T + (27.4 + 30.4i)T^{2} \)
43 \( 1 + (3.11 - 5.40i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.169 - 1.60i)T + (-45.9 + 9.77i)T^{2} \)
53 \( 1 + (-2.97 + 9.14i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.07 + 10.2i)T + (-57.7 - 12.2i)T^{2} \)
61 \( 1 + (7.68 + 1.63i)T + (55.7 + 24.8i)T^{2} \)
67 \( 1 + (-4.78 - 8.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.71 - 5.29i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-2.61 - 1.90i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-6.33 - 7.03i)T + (-8.25 + 78.5i)T^{2} \)
83 \( 1 + (-0.692 - 0.147i)T + (75.8 + 33.7i)T^{2} \)
89 \( 1 + 0.527T + 89T^{2} \)
97 \( 1 + (9.39 + 10.4i)T + (-10.1 + 96.4i)T^{2} \)
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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

−9.751952053432800012981679691316, −9.373815894490979451416012090388, −8.342248062690843574670602609343, −7.31870092956907243729565343913, −6.40276878907232280614572636694, −5.51823800859982978889459485870, −4.82851494812782669911539756118, −4.03228018569542116856540312999, −1.92658839615845194032066616976, −1.21328675903441364561798173843, 1.97424068157266204135735083077, 2.57071487052646845551532726455, 3.90269721317006412247022861267, 4.76542990582229210870488399764, 5.95360066466668744451643455652, 6.84682794029720063400958980040, 7.64941102970999729201254667860, 8.681643262727782590236813970584, 9.272218013795379737856550976622, 10.67147417001238676853531563890

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