Properties

Label 2-896-56.27-c1-0-4
Degree $2$
Conductor $896$
Sign $-0.997 + 0.0716i$
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  + 2.73i·3-s + 0.732·5-s + (2 + 1.73i)7-s − 4.46·9-s − 5.46·11-s − 4.73·13-s + 2i·15-s − 4i·17-s + 1.26i·19-s + (−4.73 + 5.46i)21-s + 5.46i·23-s − 4.46·25-s − 3.99i·27-s + 6.92i·29-s + 6.92·31-s + ⋯
L(s)  = 1  + 1.57i·3-s + 0.327·5-s + (0.755 + 0.654i)7-s − 1.48·9-s − 1.64·11-s − 1.31·13-s + 0.516i·15-s − 0.970i·17-s + 0.290i·19-s + (−1.03 + 1.19i)21-s + 1.13i·23-s − 0.892·25-s − 0.769i·27-s + 1.28i·29-s + 1.24·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0716i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0716i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $-0.997 + 0.0716i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ -0.997 + 0.0716i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0371771 - 1.03695i\)
\(L(\frac12)\) \(\approx\) \(0.0371771 - 1.03695i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 1.73i)T \)
good3 \( 1 - 2.73iT - 3T^{2} \)
5 \( 1 - 0.732T + 5T^{2} \)
11 \( 1 + 5.46T + 11T^{2} \)
13 \( 1 + 4.73T + 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 1.26iT - 19T^{2} \)
23 \( 1 - 5.46iT - 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 6.92T + 31T^{2} \)
37 \( 1 + 4iT - 37T^{2} \)
41 \( 1 - 2.92iT - 41T^{2} \)
43 \( 1 - 2.53T + 43T^{2} \)
47 \( 1 + 6.92T + 47T^{2} \)
53 \( 1 - 6.92iT - 53T^{2} \)
59 \( 1 + 3.80iT - 59T^{2} \)
61 \( 1 - 11.6T + 61T^{2} \)
67 \( 1 + 2.53T + 67T^{2} \)
71 \( 1 - 4.53iT - 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + 3.46iT - 79T^{2} \)
83 \( 1 - 10.7iT - 83T^{2} \)
89 \( 1 + 14.9iT - 89T^{2} \)
97 \( 1 - 12iT - 97T^{2} \)
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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

−10.32833209614442576848696006479, −9.798044887688660029322397730366, −9.148818371430860116604126373328, −8.124719821530397207675546763229, −7.38232468742527731645632474376, −5.70067820394470215721721253687, −5.12448716454450108178972077456, −4.63742678297712441093110923420, −3.18364242530950448584662721867, −2.30632300220758869695322856539, 0.45624222994416594933553733501, 1.97771810361146053890213385609, 2.62089834895025186585259445390, 4.46107299867152544962790831716, 5.41316743239096358681565202596, 6.40750983352129277576636035264, 7.23175189388237959629776825590, 8.060488107230734494989317426019, 8.226722941868798080185300660938, 9.921711786696231319513350521055

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