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Properties

Label 2-1456-91.90-c0-0-1
Degree $2$
Conductor $1456$
Sign $1$
Analytic cond. $0.726638$
Root an. cond. $0.852431$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s + 9-s − 13-s − 19-s + 23-s − 29-s − 31-s + 35-s − 2·41-s + 43-s + 45-s − 47-s + 49-s − 53-s + 2·59-s + 63-s − 65-s + 73-s + 79-s + 81-s − 83-s + 89-s − 91-s − 95-s + 97-s − 2·107-s + ⋯
L(s)  = 1  + 5-s + 7-s + 9-s − 13-s − 19-s + 23-s − 29-s − 31-s + 35-s − 2·41-s + 43-s + 45-s − 47-s + 49-s − 53-s + 2·59-s + 63-s − 65-s + 73-s + 79-s + 81-s − 83-s + 89-s − 91-s − 95-s + 97-s − 2·107-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1456\)    =    \(2^{4} \cdot 7 \cdot 13\)
Sign: $1$
Analytic conductor: \(0.726638\)
Root analytic conductor: \(0.852431\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1456} (545, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1456,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.439031654\)
\(L(\frac12)\) \(\approx\) \(1.439031654\)
\(L(1)\) 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 \( 1 - T \)
13 \( 1 + T \)
good3 \( ( 1 - T )( 1 + T ) \)
5 \( 1 - T + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T + T^{2} \)
23 \( 1 - T + T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 + T )^{2} \)
43 \( 1 - T + T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 + T + T^{2} \)
59 \( ( 1 - T )^{2} \)
61 \( ( 1 - T )( 1 + T ) \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T + T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 - T + T^{2} \)
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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

−9.703674470727793435140502330251, −9.062288147899012868068054316289, −8.074247124213387231062310481169, −7.25050393712006389636974250894, −6.55325330003020135667335241417, −5.37547007806551581758585711241, −4.87553420311377906844374555919, −3.81985912675114700185455249882, −2.28380745354437830918167900351, −1.61049141033235484578455353510, 1.61049141033235484578455353510, 2.28380745354437830918167900351, 3.81985912675114700185455249882, 4.87553420311377906844374555919, 5.37547007806551581758585711241, 6.55325330003020135667335241417, 7.25050393712006389636974250894, 8.074247124213387231062310481169, 9.062288147899012868068054316289, 9.703674470727793435140502330251

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