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Properties

Label 2-896-56.3-c1-0-9
Degree $2$
Conductor $896$
Sign $0.766 + 0.642i$
Analytic cond. $7.15459$
Root an. cond. $2.67480$
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  + (−1.54 − 0.893i)3-s + (−0.581 − 1.00i)5-s + (2.23 + 1.41i)7-s + (0.0963 + 0.166i)9-s + (−1.13 + 1.96i)11-s + 1.57·13-s + 2.07i·15-s + (5.96 + 3.44i)17-s + (−0.671 + 0.387i)19-s + (−2.20 − 4.18i)21-s + (−1.11 + 0.643i)23-s + (1.82 − 3.15i)25-s + 5.01i·27-s − 8.38i·29-s + (−0.554 + 0.960i)31-s + ⋯
L(s)  = 1  + (−0.893 − 0.515i)3-s + (−0.260 − 0.450i)5-s + (0.845 + 0.533i)7-s + (0.0321 + 0.0556i)9-s + (−0.342 + 0.593i)11-s + 0.438·13-s + 0.536i·15-s + (1.44 + 0.834i)17-s + (−0.154 + 0.0889i)19-s + (−0.480 − 0.912i)21-s + (−0.232 + 0.134i)23-s + (0.364 − 0.631i)25-s + 0.965i·27-s − 1.55i·29-s + (−0.0995 + 0.172i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11595 - 0.405940i\)
\(L(\frac12)\) \(\approx\) \(1.11595 - 0.405940i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2.23 - 1.41i)T \)
good3 \( 1 + (1.54 + 0.893i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.581 + 1.00i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.13 - 1.96i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.57T + 13T^{2} \)
17 \( 1 + (-5.96 - 3.44i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.671 - 0.387i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.11 - 0.643i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.38iT - 29T^{2} \)
31 \( 1 + (0.554 - 0.960i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.84 + 3.37i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 1.99iT - 41T^{2} \)
43 \( 1 - 5.64T + 43T^{2} \)
47 \( 1 + (2.90 + 5.03i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.317 - 0.183i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.6 - 6.14i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.93 + 6.81i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.01 + 6.94i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.7iT - 71T^{2} \)
73 \( 1 + (-9.70 - 5.60i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-11.6 + 6.72i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.15iT - 83T^{2} \)
89 \( 1 + (2.53 - 1.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 0.333iT - 97T^{2} \)
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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

−10.15947792729311427203443974359, −9.122994873167137171584615182478, −8.117214851135408941530844751188, −7.66185843386657338385019718168, −6.36117201703769287986147258009, −5.70948685873341079716378405757, −4.90926409279679202015422033974, −3.80589657510387802679936575360, −2.14159457348248474000485204316, −0.878986143204988167137844570532, 1.02590356353491396486376734936, 2.90068899846103309749193665834, 4.00390850051316100094885739654, 5.08968200658683184424378144445, 5.59335439630373618421223937347, 6.77168728235323142151888268066, 7.69594548217332218134362273084, 8.373579613549466505761893564838, 9.614804969281063271088795625214, 10.45876965685807276666036026156

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