Properties

Label 2-56-1.1-c1-0-1
Degree $2$
Conductor $56$
Sign $1$
Analytic cond. $0.447162$
Root an. cond. $0.668701$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·5-s + 7-s + 9-s − 8·15-s − 2·17-s − 2·19-s + 2·21-s + 8·23-s + 11·25-s − 4·27-s + 2·29-s + 4·31-s − 4·35-s − 6·37-s − 2·41-s + 8·43-s − 4·45-s − 4·47-s + 49-s − 4·51-s − 10·53-s − 4·57-s + 6·59-s + 4·61-s + 63-s − 12·67-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s + 0.377·7-s + 1/3·9-s − 2.06·15-s − 0.485·17-s − 0.458·19-s + 0.436·21-s + 1.66·23-s + 11/5·25-s − 0.769·27-s + 0.371·29-s + 0.718·31-s − 0.676·35-s − 0.986·37-s − 0.312·41-s + 1.21·43-s − 0.596·45-s − 0.583·47-s + 1/7·49-s − 0.560·51-s − 1.37·53-s − 0.529·57-s + 0.781·59-s + 0.512·61-s + 0.125·63-s − 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $1$
Analytic conductor: \(0.447162\)
Root analytic conductor: \(0.668701\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{56} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 56,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9480191156\)
\(L(\frac12)\) \(\approx\) \(0.9480191156\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 + 4 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−15.16780114648984000281229819869, −14.47141648264651804115718476828, −13.10747313979957160372604181414, −11.85083361056233693149695469963, −10.85093898417625194511634818884, −8.928329697232844766585907221767, −8.188057214509100026281306554311, −7.14438434722230698367292292955, −4.50185552371175935345657331179, −3.14137792992390816749783057891, 3.14137792992390816749783057891, 4.50185552371175935345657331179, 7.14438434722230698367292292955, 8.188057214509100026281306554311, 8.928329697232844766585907221767, 10.85093898417625194511634818884, 11.85083361056233693149695469963, 13.10747313979957160372604181414, 14.47141648264651804115718476828, 15.16780114648984000281229819869

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