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Properties

Label 2-896-16.13-c1-0-19
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: $\chi_{896} (673, \cdot )$
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

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

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