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Properties

Label 2-320-20.19-c0-0-0
Degree $2$
Conductor $320$
Sign $1$
Analytic cond. $0.159700$
Root an. cond. $0.399625$
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 − 9-s + 25-s − 2·29-s − 2·41-s − 45-s − 49-s + 2·61-s + 81-s + 2·89-s − 2·101-s + 2·109-s + ⋯
L(s)  = 1  + 5-s − 9-s + 25-s − 2·29-s − 2·41-s − 45-s − 49-s + 2·61-s + 81-s + 2·89-s − 2·101-s + 2·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(320\)    =    \(2^{6} \cdot 5\)
Sign: $1$
Analytic conductor: \(0.159700\)
Root analytic conductor: \(0.399625\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{320} (319, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 320,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8504405167\)
\(L(\frac12)\) \(\approx\) \(0.8504405167\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
good3 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( ( 1 - T )( 1 + T ) \)
23 \( 1 + T^{2} \)
29 \( ( 1 + T )^{2} \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 + T )^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 - T )^{2} \)
67 \( 1 + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )( 1 + T ) \)
83 \( 1 + T^{2} \)
89 \( ( 1 - T )^{2} \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−11.75260967574454195731534848000, −10.95046023710483770264634399978, −9.939604888849185529889975447605, −9.118094868217868109134717580806, −8.223299900747954342820438618786, −6.91232893156724944378745473872, −5.87928792951245296487148572258, −5.09929721695917834043957298071, −3.39936575199064474259124641699, −2.03689211445596801495590541896, 2.03689211445596801495590541896, 3.39936575199064474259124641699, 5.09929721695917834043957298071, 5.87928792951245296487148572258, 6.91232893156724944378745473872, 8.223299900747954342820438618786, 9.118094868217868109134717580806, 9.939604888849185529889975447605, 10.95046023710483770264634399978, 11.75260967574454195731534848000

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