↑

Properties

Label 2-896-896.59-c1-0-15
Degree $2$
Conductor $896$
Sign $0.999 + 0.0155i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.848 + 1.13i)2-s + (−2.14 − 2.28i)3-s + (−0.559 − 1.92i)4-s + (−0.811 − 1.79i)5-s + (4.41 − 0.482i)6-s + (0.0143 − 2.64i)7-s + (2.64 + 0.996i)8-s + (−0.447 + 6.82i)9-s + (2.71 + 0.601i)10-s + (2.17 + 3.49i)11-s + (−3.19 + 5.39i)12-s + (0.461 + 4.68i)13-s + (2.98 + 2.26i)14-s + (−2.36 + 5.69i)15-s + (−3.37 + 2.14i)16-s + (2.40 + 0.316i)17-s + ⋯
L(s)  = 1  + (−0.600 + 0.799i)2-s + (−1.23 − 1.32i)3-s + (−0.279 − 0.960i)4-s + (−0.362 − 0.800i)5-s + (1.80 − 0.197i)6-s + (0.00544 − 0.999i)7-s + (0.935 + 0.352i)8-s + (−0.149 + 2.27i)9-s + (0.858 + 0.190i)10-s + (0.654 + 1.05i)11-s + (−0.922 + 1.55i)12-s + (0.127 + 1.29i)13-s + (0.796 + 0.604i)14-s + (−0.609 + 1.47i)15-s + (−0.843 + 0.537i)16-s + (0.583 + 0.0768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0155i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.999 + 0.0155i$
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: \((2,\ 896,\ (\ :1/2),\ 0.999 + 0.0155i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.632225 - 0.00491272i\)
\(L(\frac12)\) \(\approx\) \(0.632225 - 0.00491272i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.848 - 1.13i)T \)
7 \( 1 + (-0.0143 + 2.64i)T \)
good3 \( 1 + (2.14 + 2.28i)T + (-0.196 + 2.99i)T^{2} \)
5 \( 1 + (0.811 + 1.79i)T + (-3.29 + 3.75i)T^{2} \)
11 \( 1 + (-2.17 - 3.49i)T + (-4.86 + 9.86i)T^{2} \)
13 \( 1 + (-0.461 - 4.68i)T + (-12.7 + 2.53i)T^{2} \)
17 \( 1 + (-2.40 - 0.316i)T + (16.4 + 4.39i)T^{2} \)
19 \( 1 + (-0.0519 + 0.314i)T + (-17.9 - 6.10i)T^{2} \)
23 \( 1 + (-1.92 - 2.18i)T + (-3.00 + 22.8i)T^{2} \)
29 \( 1 + (-2.62 - 4.91i)T + (-16.1 + 24.1i)T^{2} \)
31 \( 1 + (-6.35 + 1.70i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.01 - 7.99i)T + (-27.8 - 24.3i)T^{2} \)
41 \( 1 + (-0.201 - 1.01i)T + (-37.8 + 15.6i)T^{2} \)
43 \( 1 + (-0.267 + 0.0810i)T + (35.7 - 23.8i)T^{2} \)
47 \( 1 + (-4.73 + 6.17i)T + (-12.1 - 45.3i)T^{2} \)
53 \( 1 + (-1.66 - 2.67i)T + (-23.4 + 47.5i)T^{2} \)
59 \( 1 + (6.36 - 8.88i)T + (-18.9 - 55.8i)T^{2} \)
61 \( 1 + (10.9 - 2.55i)T + (54.7 - 26.9i)T^{2} \)
67 \( 1 + (2.38 + 2.54i)T + (-4.38 + 66.8i)T^{2} \)
71 \( 1 + (2.54 + 3.80i)T + (-27.1 + 65.5i)T^{2} \)
73 \( 1 + (1.13 - 0.559i)T + (44.4 - 57.9i)T^{2} \)
79 \( 1 + (-1.89 - 14.4i)T + (-76.3 + 20.4i)T^{2} \)
83 \( 1 + (-8.45 + 10.3i)T + (-16.1 - 81.4i)T^{2} \)
89 \( 1 + (-4.68 - 13.7i)T + (-70.6 + 54.1i)T^{2} \)
97 \( 1 + (-6.34 + 6.34i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.13997235797911977165139659473, −9.108182299117926495063666562205, −8.137486963864181274301802778736, −7.29546088858128711988918532791, −6.85986966864893595614640344624, −6.16022200631232201807590961300, −4.91123975198523388110786888270, −4.43339058725470350671514870291, −1.60475509435105336905479709829, −0.974165028802309742064074361534, 0.62244107850407688730459981541, 2.96570549262373515369614518066, 3.51125653195417856099586539104, 4.68825374735375201418976643298, 5.71110437628815508013765755332, 6.43178816733310527842547748019, 7.82819440271581009528108702028, 8.815061210641026511197400820124, 9.477144186611855868201281893038, 10.51782336236261829929606149333

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