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Properties

Label 2-896-56.13-c0-0-3
Degree $2$
Conductor $896$
Sign $1$
Analytic cond. $0.447162$
Root an. cond. $0.668701$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s + 1.41·5-s − 7-s + 1.00·9-s − 1.41·13-s + 2.00·15-s − 1.41·19-s − 1.41·21-s + 1.00·25-s − 1.41·35-s − 2.00·39-s + 1.41·45-s + 49-s − 2.00·57-s − 1.41·59-s + 1.41·61-s − 1.00·63-s − 2.00·65-s + 2·71-s + 1.41·75-s + 2·79-s − 0.999·81-s + 1.41·83-s + 1.41·91-s − 2.00·95-s − 1.41·101-s − 2.00·105-s + ⋯
L(s)  = 1  + 1.41·3-s + 1.41·5-s − 7-s + 1.00·9-s − 1.41·13-s + 2.00·15-s − 1.41·19-s − 1.41·21-s + 1.00·25-s − 1.41·35-s − 2.00·39-s + 1.41·45-s + 49-s − 2.00·57-s − 1.41·59-s + 1.41·61-s − 1.00·63-s − 2.00·65-s + 2·71-s + 1.41·75-s + 2·79-s − 0.999·81-s + 1.41·83-s + 1.41·91-s − 2.00·95-s − 1.41·101-s − 2.00·105-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (321, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 896,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.609862415\)
\(L(\frac12)\) \(\approx\) \(1.609862415\)
\(L(1)\) 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 \( 1 + T \)
good3 \( 1 - 1.41T + T^{2} \)
5 \( 1 - 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.41T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - 1.41T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 2T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 2T + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 - T^{2} \)
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

−9.905047453160838557555947178868, −9.527308881005127497001650019730, −8.874171836849939427887581816672, −7.918725979961247470488906846370, −6.88928402697298387294447091967, −6.15758523147932605603450862004, −5.01908424709103718454780659728, −3.73845839731350988791793931379, −2.59903864581965609251739485447, −2.11120570570552100753675427989, 2.11120570570552100753675427989, 2.59903864581965609251739485447, 3.73845839731350988791793931379, 5.01908424709103718454780659728, 6.15758523147932605603450862004, 6.88928402697298387294447091967, 7.918725979961247470488906846370, 8.874171836849939427887581816672, 9.527308881005127497001650019730, 9.905047453160838557555947178868

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