Properties

Label 2-896-56.27-c1-0-23
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  + 0.732i·3-s + 2.73·5-s + (2 − 1.73i)7-s + 2.46·9-s − 1.46·11-s + 1.26·13-s + 2i·15-s − 4i·17-s − 4.73i·19-s + (1.26 + 1.46i)21-s − 1.46i·23-s + 2.46·25-s + 4i·27-s + 6.92i·29-s − 6.92·31-s + ⋯
L(s)  = 1  + 0.422i·3-s + 1.22·5-s + (0.755 − 0.654i)7-s + 0.821·9-s − 0.441·11-s + 0.351·13-s + 0.516i·15-s − 0.970i·17-s − 1.08i·19-s + (0.276 + 0.319i)21-s − 0.305i·23-s + 0.492·25-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

\(L(1)\) \(\approx\) \(2.19874 - 0.0788298i\)
\(L(\frac12)\) \(\approx\) \(2.19874 - 0.0788298i\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.732iT - 3T^{2} \)
5 \( 1 - 2.73T + 5T^{2} \)
11 \( 1 + 1.46T + 11T^{2} \)
13 \( 1 - 1.26T + 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 + 4.73iT - 19T^{2} \)
23 \( 1 + 1.46iT - 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 10.9iT - 41T^{2} \)
43 \( 1 + 9.46T + 43T^{2} \)
47 \( 1 - 6.92T + 47T^{2} \)
53 \( 1 - 6.92iT - 53T^{2} \)
59 \( 1 - 14.1iT - 59T^{2} \)
61 \( 1 - 5.66T + 61T^{2} \)
67 \( 1 - 9.46T + 67T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 - 3.46iT - 79T^{2} \)
83 \( 1 + 7.26iT - 83T^{2} \)
89 \( 1 + 1.07iT - 89T^{2} \)
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

−10.24302320667264192152045175207, −9.323445116243370887207728998323, −8.675177423066362789548391516358, −7.32709335087679600290435756314, −6.87350587358488049959631539914, −5.46086489129919894735645032491, −4.94999344189708963318868152447, −3.86756551138795047522610459763, −2.45167436813985850173032325578, −1.26493397192796810508723456933, 1.61340284455675737489073043688, 2.12368546178447106337276691732, 3.76634206770970039153092937335, 5.03633897638830039365118085678, 5.85600286798216815119185839286, 6.48558041224434865201750243035, 7.76749136949187647746303010460, 8.291757319853248581563968770567, 9.446728875363881841466817406782, 10.02054104066198488401605234823

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