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Properties

Label 2-896-8.5-c1-0-13
Degree $2$
Conductor $896$
Sign $1$
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  + 1.08i·3-s − 1.08i·5-s − 7-s + 1.82·9-s − 5.22i·11-s + 6.30i·13-s + 1.17·15-s + 3.65·17-s − 4.14i·19-s − 1.08i·21-s − 1.17·23-s + 3.82·25-s + 5.22i·27-s − 8.28i·29-s + 5.65·31-s + ⋯
L(s)  = 1  + 0.624i·3-s − 0.484i·5-s − 0.377·7-s + 0.609·9-s − 1.57i·11-s + 1.74i·13-s + 0.302·15-s + 0.886·17-s − 0.950i·19-s − 0.236i·21-s − 0.244·23-s + 0.765·25-s + 1.00i·27-s − 1.53i·29-s + 1.01·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $1$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63752\)
\(L(\frac12)\) \(\approx\) \(1.63752\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good3 \( 1 - 1.08iT - 3T^{2} \)
5 \( 1 + 1.08iT - 5T^{2} \)
11 \( 1 + 5.22iT - 11T^{2} \)
13 \( 1 - 6.30iT - 13T^{2} \)
17 \( 1 - 3.65T + 17T^{2} \)
19 \( 1 + 4.14iT - 19T^{2} \)
23 \( 1 + 1.17T + 23T^{2} \)
29 \( 1 + 8.28iT - 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + 2.16iT - 37T^{2} \)
41 \( 1 - 7.65T + 41T^{2} \)
43 \( 1 - 5.22iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 - 4.32iT - 53T^{2} \)
59 \( 1 + 6.30iT - 59T^{2} \)
61 \( 1 - 7.20iT - 61T^{2} \)
67 \( 1 - 7.39iT - 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 0.343T + 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 + 5.41iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−9.989140511287009244354355653196, −9.246486752866700899245840494681, −8.731372102233984277719821839871, −7.61694612204372121044984870151, −6.57541740603830524177403390152, −5.76694528978135966106523776545, −4.58121578400067069975973221691, −3.96903067483209967287134520277, −2.73586866346125114337555037798, −1.00603784400427532290366559574, 1.20216301084953963803582209224, 2.54655482498280317382189571805, 3.61024147735389999189844250735, 4.86674656817961731328386399614, 5.88253989446817293065956727907, 6.88485917240661119438985936688, 7.48640582211781263320515150529, 8.140338995648039637841101440132, 9.472070271214050112182526675492, 10.32876765713086661865512205290

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