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Properties

Label 2-896-7.4-c1-0-3
Degree $2$
Conductor $896$
Sign $-0.961 + 0.274i$
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  + (−1.07 + 1.85i)3-s + (1.93 + 3.35i)5-s + (−0.886 − 2.49i)7-s + (−0.795 − 1.37i)9-s + (−1.92 + 3.32i)11-s − 4.69·13-s − 8.30·15-s + (−0.257 + 0.445i)17-s + (−1.95 − 3.39i)19-s + (5.57 + 1.02i)21-s + (0.979 + 1.69i)23-s + (−5.02 + 8.69i)25-s − 3.02·27-s + 9.34·29-s + (0.723 − 1.25i)31-s + ⋯
L(s)  = 1  + (−0.618 + 1.07i)3-s + (0.867 + 1.50i)5-s + (−0.335 − 0.942i)7-s + (−0.265 − 0.459i)9-s + (−0.579 + 1.00i)11-s − 1.30·13-s − 2.14·15-s + (−0.0624 + 0.108i)17-s + (−0.449 − 0.778i)19-s + (1.21 + 0.223i)21-s + (0.204 + 0.353i)23-s + (−1.00 + 1.73i)25-s − 0.581·27-s + 1.73·29-s + (0.129 − 0.225i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.961 + 0.274i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.961 + 0.274i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -0.961 + 0.274i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.109555 - 0.781931i\)
\(L(\frac12)\) \(\approx\) \(0.109555 - 0.781931i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.886 + 2.49i)T \)
good3 \( 1 + (1.07 - 1.85i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.93 - 3.35i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.92 - 3.32i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.69T + 13T^{2} \)
17 \( 1 + (0.257 - 0.445i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.95 + 3.39i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.979 - 1.69i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.34T + 29T^{2} \)
31 \( 1 + (-0.723 + 1.25i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.32 - 4.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.06T + 41T^{2} \)
43 \( 1 + 5.75T + 43T^{2} \)
47 \( 1 + (-1.68 - 2.92i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.224 + 0.389i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.80 + 3.13i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.30 + 12.6i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.25 - 7.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-5.74 + 9.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.44 - 5.97i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 + (-3.78 - 6.56i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 3.59T + 97T^{2} \)
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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

−10.43211758279471673526956878223, −9.935902567372975750570276778328, −9.549102128610269784099332272782, −7.80469155497944730323395899643, −6.92247858633002716613407960579, −6.43992267622040248476306055807, −5.10905391587998476803494474642, −4.54546442800085821548532539542, −3.23607251074854642606090223078, −2.26156730334702737109996432322, 0.38426082132328570524326185496, 1.67493827325431533515293728142, 2.73288098556057364991886141498, 4.67243938400816613416441688247, 5.50878355890679619153426515544, 5.98729838230946066135825410255, 6.91108295140244884373221177098, 8.190896504962649877174396541604, 8.655083248922588518309164552754, 9.613288141765459841588065010732

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