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Properties

Label 2-891-99.58-c1-0-31
Degree $2$
Conductor $891$
Sign $0.737 - 0.675i$
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  + (2.09 + 0.934i)2-s + (2.19 + 2.43i)4-s + (1.55 − 0.690i)5-s + (−0.203 + 0.0431i)7-s + (0.910 + 2.80i)8-s + 3.90·10-s + (2.04 − 2.60i)11-s + (−0.144 − 1.37i)13-s + (−0.466 − 0.0992i)14-s + (−0.0217 + 0.206i)16-s + (2.97 + 2.15i)17-s + (2.50 + 7.70i)19-s + (5.08 + 2.26i)20-s + (6.73 − 3.56i)22-s + (−1.99 + 3.45i)23-s + ⋯
L(s)  = 1  + (1.48 + 0.660i)2-s + (1.09 + 1.21i)4-s + (0.693 − 0.308i)5-s + (−0.0767 + 0.0163i)7-s + (0.321 + 0.990i)8-s + 1.23·10-s + (0.617 − 0.786i)11-s + (−0.0400 − 0.381i)13-s + (−0.124 − 0.0265i)14-s + (−0.00542 + 0.0516i)16-s + (0.720 + 0.523i)17-s + (0.574 + 1.76i)19-s + (1.13 + 0.506i)20-s + (1.43 − 0.759i)22-s + (−0.416 + 0.721i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.737 - 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(891\)    =    \(3^{4} \cdot 11\)
Sign: $0.737 - 0.675i$
Analytic conductor: \(7.11467\)
Root analytic conductor: \(2.66733\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{891} (190, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 891,\ (\ :1/2),\ 0.737 - 0.675i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.74667 + 1.45669i\)
\(L(\frac12)\) \(\approx\) \(3.74667 + 1.45669i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 + 2.60i)T \)
good2 \( 1 + (-2.09 - 0.934i)T + (1.33 + 1.48i)T^{2} \)
5 \( 1 + (-1.55 + 0.690i)T + (3.34 - 3.71i)T^{2} \)
7 \( 1 + (0.203 - 0.0431i)T + (6.39 - 2.84i)T^{2} \)
13 \( 1 + (0.144 + 1.37i)T + (-12.7 + 2.70i)T^{2} \)
17 \( 1 + (-2.97 - 2.15i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-2.50 - 7.70i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.99 - 3.45i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.530 - 0.112i)T + (26.4 - 11.7i)T^{2} \)
31 \( 1 + (1.02 + 9.77i)T + (-30.3 + 6.44i)T^{2} \)
37 \( 1 + (0.381 - 1.17i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (9.13 + 1.94i)T + (37.4 + 16.6i)T^{2} \)
43 \( 1 + (3.96 + 6.86i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.25 - 5.83i)T + (-4.91 - 46.7i)T^{2} \)
53 \( 1 + (5.30 - 3.85i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-5.26 - 5.84i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (-0.145 + 1.38i)T + (-59.6 - 12.6i)T^{2} \)
67 \( 1 + (-2.92 + 5.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-10.3 - 7.49i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.36 + 7.27i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (5.30 + 2.36i)T + (52.8 + 58.7i)T^{2} \)
83 \( 1 + (-1.45 + 13.7i)T + (-81.1 - 17.2i)T^{2} \)
89 \( 1 + 6.19T + 89T^{2} \)
97 \( 1 + (6.41 + 2.85i)T + (64.9 + 72.0i)T^{2} \)
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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

−10.10967024684845905861342676053, −9.502238422272909581961902565532, −8.198303965349340324822779579843, −7.54599599710368279524250782657, −6.26598472196005653006616284984, −5.83472643061899524898232460934, −5.20398337179677778083050404602, −3.85670447607201129126650059095, −3.33634410245590151166350592195, −1.65555379681884070845652021226, 1.64873074716557662719826053874, 2.66765120688604185935686543274, 3.57950605457240475530221167268, 4.79036417587050639679405463928, 5.22464122100216280999376401174, 6.59822860697523277573296913829, 6.81927020877516601420296670426, 8.415699260225294192664390608482, 9.614124654046048214090131294665, 10.08894674811143812754101736841

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