Properties

Label 2-1456-1.1-c1-0-25
Degree $2$
Conductor $1456$
Sign $-1$
Analytic cond. $11.6262$
Root an. cond. $3.40972$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 7-s − 3·9-s + 6·11-s − 13-s + 4·17-s − 5·19-s − 3·23-s + 4·25-s − 5·29-s + 3·31-s − 3·35-s − 4·37-s − 6·41-s + 43-s + 9·45-s − 7·47-s + 49-s − 9·53-s − 18·55-s − 8·59-s − 10·61-s − 3·63-s + 3·65-s + 6·67-s + 8·71-s − 13·73-s + ⋯
L(s)  = 1  − 1.34·5-s + 0.377·7-s − 9-s + 1.80·11-s − 0.277·13-s + 0.970·17-s − 1.14·19-s − 0.625·23-s + 4/5·25-s − 0.928·29-s + 0.538·31-s − 0.507·35-s − 0.657·37-s − 0.937·41-s + 0.152·43-s + 1.34·45-s − 1.02·47-s + 1/7·49-s − 1.23·53-s − 2.42·55-s − 1.04·59-s − 1.28·61-s − 0.377·63-s + 0.372·65-s + 0.733·67-s + 0.949·71-s − 1.52·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $-1$
Analytic conductor: \(11.6262\)
Root analytic conductor: \(3.40972\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1456} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1456,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 + T \)
good3 \( 1 + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 5 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 6 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 15 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−8.920719672465111957084037828236, −8.306653899443735547493881784402, −7.65856427227425769928738953272, −6.69324481208496363874135514467, −5.91107970337180073579003850586, −4.71772169836810783436044308640, −3.91316951599567757387300319071, −3.20361346457630305584783093767, −1.62321334030766970947302300597, 0, 1.62321334030766970947302300597, 3.20361346457630305584783093767, 3.91316951599567757387300319071, 4.71772169836810783436044308640, 5.91107970337180073579003850586, 6.69324481208496363874135514467, 7.65856427227425769928738953272, 8.306653899443735547493881784402, 8.920719672465111957084037828236

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