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Properties

Label 2-896-16.5-c1-0-7
Degree $2$
Conductor $896$
Sign $-0.103 - 0.994i$
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  + (−0.631 + 0.631i)3-s + (2.34 + 2.34i)5-s + i·7-s + 2.20i·9-s + (2.18 + 2.18i)11-s + (4.03 − 4.03i)13-s − 2.95·15-s + 0.347·17-s + (−4.26 + 4.26i)19-s + (−0.631 − 0.631i)21-s − 6.23i·23-s + 5.97i·25-s + (−3.28 − 3.28i)27-s + (−1.21 + 1.21i)29-s − 1.26·31-s + ⋯
L(s)  = 1  + (−0.364 + 0.364i)3-s + (1.04 + 1.04i)5-s + 0.377i·7-s + 0.734i·9-s + (0.658 + 0.658i)11-s + (1.11 − 1.11i)13-s − 0.763·15-s + 0.0843·17-s + (−0.978 + 0.978i)19-s + (−0.137 − 0.137i)21-s − 1.30i·23-s + 1.19i·25-s + (−0.632 − 0.632i)27-s + (−0.226 + 0.226i)29-s − 0.226·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.103 - 0.994i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (225, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -0.103 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14343 + 1.26905i\)
\(L(\frac12)\) \(\approx\) \(1.14343 + 1.26905i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 + (0.631 - 0.631i)T - 3iT^{2} \)
5 \( 1 + (-2.34 - 2.34i)T + 5iT^{2} \)
11 \( 1 + (-2.18 - 2.18i)T + 11iT^{2} \)
13 \( 1 + (-4.03 + 4.03i)T - 13iT^{2} \)
17 \( 1 - 0.347T + 17T^{2} \)
19 \( 1 + (4.26 - 4.26i)T - 19iT^{2} \)
23 \( 1 + 6.23iT - 23T^{2} \)
29 \( 1 + (1.21 - 1.21i)T - 29iT^{2} \)
31 \( 1 + 1.26T + 31T^{2} \)
37 \( 1 + (-6.42 - 6.42i)T + 37iT^{2} \)
41 \( 1 - 2.68iT - 41T^{2} \)
43 \( 1 + (4.05 + 4.05i)T + 43iT^{2} \)
47 \( 1 - 4.64T + 47T^{2} \)
53 \( 1 + (8.44 + 8.44i)T + 53iT^{2} \)
59 \( 1 + (-5.17 - 5.17i)T + 59iT^{2} \)
61 \( 1 + (-0.00533 + 0.00533i)T - 61iT^{2} \)
67 \( 1 + (3.02 - 3.02i)T - 67iT^{2} \)
71 \( 1 + 0.828iT - 71T^{2} \)
73 \( 1 - 6.25iT - 73T^{2} \)
79 \( 1 + 0.755T + 79T^{2} \)
83 \( 1 + (-3.66 + 3.66i)T - 83iT^{2} \)
89 \( 1 + 6.24iT - 89T^{2} \)
97 \( 1 - 2.18T + 97T^{2} \)
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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

−10.37717177894425682750879766416, −9.844005732696225724948039107058, −8.680379782556700152766948805314, −7.86572231529235415987163992374, −6.57279330865490398638859199926, −6.11072317715412374806023732585, −5.24334998247330062667319867253, −4.05141505757157336934933008080, −2.79996794136130444385858469669, −1.78449218816390114012426506905, 0.920902427067325158886176449702, 1.81422483572803884496974678535, 3.62807393334088854063232445277, 4.54640242844829020990989689137, 5.82793043882619835243438859292, 6.21237722253291915726846822779, 7.12879198949755498022735824174, 8.471886463906126885084468389974, 9.253140657007494847924670718238, 9.464484066072629401795537312775

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