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Properties

Label 2-896-896.59-c1-0-124
Degree $2$
Conductor $896$
Sign $0.878 - 0.478i$
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  + (0.223 − 1.39i)2-s + (−2.05 − 2.19i)3-s + (−1.90 − 0.623i)4-s + (−0.906 − 2.00i)5-s + (−3.52 + 2.38i)6-s + (−0.667 − 2.56i)7-s + (−1.29 + 2.51i)8-s + (−0.396 + 6.04i)9-s + (−2.99 + 0.819i)10-s + (−2.85 − 4.59i)11-s + (2.53 + 5.45i)12-s + (0.268 + 2.72i)13-s + (−3.72 + 0.359i)14-s + (−2.53 + 6.10i)15-s + (3.22 + 2.37i)16-s + (0.579 + 0.0763i)17-s + ⋯
L(s)  = 1  + (0.157 − 0.987i)2-s + (−1.18 − 1.26i)3-s + (−0.950 − 0.311i)4-s + (−0.405 − 0.894i)5-s + (−1.43 + 0.972i)6-s + (−0.252 − 0.967i)7-s + (−0.457 + 0.888i)8-s + (−0.132 + 2.01i)9-s + (−0.947 + 0.259i)10-s + (−0.861 − 1.38i)11-s + (0.733 + 1.57i)12-s + (0.0744 + 0.755i)13-s + (−0.995 + 0.0961i)14-s + (−0.653 + 1.57i)15-s + (0.805 + 0.592i)16-s + (0.140 + 0.0185i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.878 - 0.478i$
Analytic conductor: \(7.15459\)
Root analytic conductor: \(2.67480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :1/2),\ 0.878 - 0.478i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.436993 + 0.111324i\)
\(L(\frac12)\) \(\approx\) \(0.436993 + 0.111324i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.223 + 1.39i)T \)
7 \( 1 + (0.667 + 2.56i)T \)
good3 \( 1 + (2.05 + 2.19i)T + (-0.196 + 2.99i)T^{2} \)
5 \( 1 + (0.906 + 2.00i)T + (-3.29 + 3.75i)T^{2} \)
11 \( 1 + (2.85 + 4.59i)T + (-4.86 + 9.86i)T^{2} \)
13 \( 1 + (-0.268 - 2.72i)T + (-12.7 + 2.53i)T^{2} \)
17 \( 1 + (-0.579 - 0.0763i)T + (16.4 + 4.39i)T^{2} \)
19 \( 1 + (-0.0684 + 0.414i)T + (-17.9 - 6.10i)T^{2} \)
23 \( 1 + (3.84 + 4.38i)T + (-3.00 + 22.8i)T^{2} \)
29 \( 1 + (4.07 + 7.62i)T + (-16.1 + 24.1i)T^{2} \)
31 \( 1 + (-7.12 + 1.90i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.05 + 2.79i)T + (-27.8 - 24.3i)T^{2} \)
41 \( 1 + (-0.0934 - 0.469i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-7.19 + 2.18i)T + (35.7 - 23.8i)T^{2} \)
47 \( 1 + (5.83 - 7.60i)T + (-12.1 - 45.3i)T^{2} \)
53 \( 1 + (1.32 + 2.13i)T + (-23.4 + 47.5i)T^{2} \)
59 \( 1 + (-2.09 + 2.91i)T + (-18.9 - 55.8i)T^{2} \)
61 \( 1 + (-13.2 + 3.09i)T + (54.7 - 26.9i)T^{2} \)
67 \( 1 + (6.52 + 6.96i)T + (-4.38 + 66.8i)T^{2} \)
71 \( 1 + (-8.55 - 12.8i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (-11.0 + 5.44i)T + (44.4 - 57.9i)T^{2} \)
79 \( 1 + (1.12 + 8.54i)T + (-76.3 + 20.4i)T^{2} \)
83 \( 1 + (0.335 - 0.409i)T + (-16.1 - 81.4i)T^{2} \)
89 \( 1 + (-1.57 - 4.63i)T + (-70.6 + 54.1i)T^{2} \)
97 \( 1 + (12.4 - 12.4i)T - 97iT^{2} \)
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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

−9.558519470885159056803028560067, −8.218325521992028813846248077213, −7.923270955504434682315905757100, −6.51757135924234226572868388131, −5.81803281044301942597721250148, −4.82705093017094025324915939798, −3.95782627954970310267943322914, −2.36367323444044140173131509671, −0.918496905649568388659672290516, −0.32565317481602561285876350595, 3.01589851328779060672665760798, 3.99459826522052451228039298665, 5.12334683189021366253899773394, 5.46433193638697162421791661141, 6.46132202734955940762990951250, 7.27675802424107583673587197582, 8.267479204828811537286919991387, 9.470146002304740182784415125321, 9.986320315511715793339563393810, 10.66431575639560990960148128168

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