Properties

Label 2-896-224.125-c0-0-0
Degree $2$
Conductor $896$
Sign $0.831 - 0.555i$
Analytic cond. $0.447162$
Root an. cond. $0.668701$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (0.707 − 0.292i)11-s + (1 + i)23-s + (−0.707 + 0.707i)25-s + (0.707 + 0.292i)29-s + (−0.707 − 1.70i)37-s + (1.70 − 0.707i)43-s − 1.00i·49-s + (−1.70 + 0.707i)53-s − 1.00·63-s + (−1.70 − 0.707i)67-s + (−0.292 + 0.707i)77-s − 1.41i·79-s + 1.00i·81-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)7-s + (0.707 + 0.707i)9-s + (0.707 − 0.292i)11-s + (1 + i)23-s + (−0.707 + 0.707i)25-s + (0.707 + 0.292i)29-s + (−0.707 − 1.70i)37-s + (1.70 − 0.707i)43-s − 1.00i·49-s + (−1.70 + 0.707i)53-s − 1.00·63-s + (−1.70 − 0.707i)67-s + (−0.292 + 0.707i)77-s − 1.41i·79-s + 1.00i·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.831 - 0.555i$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :0),\ 0.831 - 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.001402116\)
\(L(\frac12)\) \(\approx\) \(1.001402116\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (-0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.707 - 0.707i)T^{2} \)
11 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
13 \( 1 + (0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (0.707 + 0.707i)T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.707 - 0.707i)T^{2} \)
67 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41iT - T^{2} \)
83 \( 1 + (0.707 + 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
97 \( 1 - T^{2} \)
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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

−10.43114020442701134391521099516, −9.301182373677471663919291295137, −9.067152772628851598352349845602, −7.75023948533664506084751229972, −7.05160542580676998808925786043, −6.03269040468560780819678370674, −5.23372999870034590157920533037, −4.04899665595233899823002489118, −3.01838524610478155287104693419, −1.68642899387736928874792203941, 1.17278319926453268689170926646, 2.90114223147121109838553647662, 3.98667709605783841911901767470, 4.68245350256690288926292139439, 6.28856204829249826704379628031, 6.66499596100028099353452090499, 7.57494709329354829425291154387, 8.672870259907173974021927159507, 9.578036305130516829698202428167, 10.07714143531296779856097997375

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