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Properties

Label 2-891-11.3-c1-0-30
Degree $2$
Conductor $891$
Sign $0.876 + 0.480i$
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  + (1.25 − 0.908i)2-s + (0.120 − 0.371i)4-s + (0.478 + 0.347i)5-s + (0.223 − 0.686i)7-s + (0.768 + 2.36i)8-s + 0.915·10-s + (0.539 − 3.27i)11-s + (1.27 − 0.925i)13-s + (−0.344 − 1.06i)14-s + (3.74 + 2.72i)16-s + (3.71 + 2.69i)17-s + (0.775 + 2.38i)19-s + (0.187 − 0.135i)20-s + (−2.29 − 4.58i)22-s + 4.45·23-s + ⋯
L(s)  = 1  + (0.884 − 0.642i)2-s + (0.0603 − 0.185i)4-s + (0.214 + 0.155i)5-s + (0.0843 − 0.259i)7-s + (0.271 + 0.836i)8-s + 0.289·10-s + (0.162 − 0.986i)11-s + (0.353 − 0.256i)13-s + (−0.0921 − 0.283i)14-s + (0.936 + 0.680i)16-s + (0.901 + 0.654i)17-s + (0.177 + 0.547i)19-s + (0.0418 − 0.0303i)20-s + (−0.490 − 0.977i)22-s + 0.928·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 + 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.876 + 0.480i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (487, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.876 + 0.480i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.64380 - 0.676995i\)
\(L(\frac12)\) \(\approx\) \(2.64380 - 0.676995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.539 + 3.27i)T \)
good2 \( 1 + (-1.25 + 0.908i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-0.478 - 0.347i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.223 + 0.686i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.27 + 0.925i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.71 - 2.69i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.775 - 2.38i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4.45T + 23T^{2} \)
29 \( 1 + (2.15 - 6.63i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-7.38 + 5.36i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-0.893 + 2.75i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-0.356 - 1.09i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4.21T + 43T^{2} \)
47 \( 1 + (-0.0703 - 0.216i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.61 - 3.35i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.20 - 6.78i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.68 + 2.67i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 8.09T + 67T^{2} \)
71 \( 1 + (9.95 + 7.23i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-4.78 + 14.7i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-11.8 + 8.59i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (5.62 + 4.08i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 12.4T + 89T^{2} \)
97 \( 1 + (11.4 - 8.33i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.54816620517047097241444025541, −9.269031555326371998429338690071, −8.290715346124659668411053444389, −7.66694211992846650665736047700, −6.25011549044733910738884776425, −5.61032136410902960901463142402, −4.52095499109511898559462871075, −3.55864953283239190590214599844, −2.85512645824069858473554442624, −1.36497683572095710186890605691, 1.32427446920521030186771679076, 2.94967823196417103074075737507, 4.18321628423769518920962817088, 5.01190325046130474420062036311, 5.66186873810033566130654637728, 6.74117618281514841868942816725, 7.27245722459309495693993282380, 8.400804040991434751438092106149, 9.590080256790519535321921839253, 9.869817374759595629242485569522

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