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Properties

Label 2-896-56.3-c1-0-12
Degree $2$
Conductor $896$
Sign $-0.0902 - 0.995i$
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  + (2.80 + 1.61i)3-s + (1.53 + 2.65i)5-s + (−2.57 + 0.591i)7-s + (3.74 + 6.48i)9-s + (1.21 − 2.10i)11-s − 0.365·13-s + 9.90i·15-s + (−3.79 − 2.18i)17-s + (4.27 − 2.46i)19-s + (−8.18 − 2.51i)21-s + (−0.108 + 0.0623i)23-s + (−2.18 + 3.78i)25-s + 14.5i·27-s − 1.73i·29-s + (4.01 − 6.95i)31-s + ⋯
L(s)  = 1  + (1.61 + 0.934i)3-s + (0.684 + 1.18i)5-s + (−0.974 + 0.223i)7-s + (1.24 + 2.16i)9-s + (0.365 − 0.633i)11-s − 0.101·13-s + 2.55i·15-s + (−0.919 − 0.530i)17-s + (0.979 − 0.565i)19-s + (−1.78 − 0.549i)21-s + (−0.0225 + 0.0130i)23-s + (−0.436 + 0.756i)25-s + 2.79i·27-s − 0.321i·29-s + (0.721 − 1.24i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0902 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0902 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.0902 - 0.995i$
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: \((2,\ 896,\ (\ :1/2),\ -0.0902 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87267 + 2.05003i\)
\(L(\frac12)\) \(\approx\) \(1.87267 + 2.05003i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.57 - 0.591i)T \)
good3 \( 1 + (-2.80 - 1.61i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.53 - 2.65i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.21 + 2.10i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.365T + 13T^{2} \)
17 \( 1 + (3.79 + 2.18i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.27 + 2.46i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.108 - 0.0623i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.73iT - 29T^{2} \)
31 \( 1 + (-4.01 + 6.95i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (6.87 - 3.96i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 6.84iT - 41T^{2} \)
43 \( 1 + 2.36T + 43T^{2} \)
47 \( 1 + (0.550 + 0.954i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.2 - 6.51i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.41 + 1.39i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.45 + 11.1i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.25 + 3.90i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.46iT - 71T^{2} \)
73 \( 1 + (4.41 + 2.54i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.3 + 5.97i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.730iT - 83T^{2} \)
89 \( 1 + (9.10 - 5.25i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−10.07745402623134707976944839255, −9.457695568507825614844734682854, −9.003782573897045528091202977224, −7.938247592248636181331090967039, −6.96574271517410803031553404705, −6.17968574380631766312137525938, −4.82966532478555167958786362566, −3.60498364691996325264537206691, −2.96048389900101727452740476428, −2.30410762582374199379860424716, 1.20808205576986450404552175032, 2.11643807108085585191169570224, 3.29894285805991129438911768248, 4.25498507070331081511790509371, 5.63736444490962401789004923964, 6.80813154602827690653655045373, 7.27263556252194260658663209395, 8.580382278879872027744183469428, 8.814939409552082051967076646177, 9.642867072579129620185821922033

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