Properties

Label 2-328-328.163-c0-0-0
Degree $2$
Conductor $328$
Sign $1$
Analytic cond. $0.163693$
Root an. cond. $0.404590$
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  − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s + 25-s − 32-s + 36-s − 41-s − 2·43-s − 49-s − 50-s − 2·59-s + 64-s − 72-s − 2·73-s + 81-s + 82-s + 2·83-s + 2·86-s + 98-s + 100-s + 2·107-s − 2·113-s + 2·118-s + ⋯
L(s)  = 1  − 2-s + 4-s − 8-s + 9-s + 16-s − 18-s + 25-s − 32-s + 36-s − 41-s − 2·43-s − 49-s − 50-s − 2·59-s + 64-s − 72-s − 2·73-s + 81-s + 82-s + 2·83-s + 2·86-s + 98-s + 100-s + 2·107-s − 2·113-s + 2·118-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(328\)    =    \(2^{3} \cdot 41\)
Sign: $1$
Analytic conductor: \(0.163693\)
Root analytic conductor: \(0.404590\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{328} (163, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 328,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5632208795\)
\(L(\frac12)\) \(\approx\) \(0.5632208795\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−11.66752597725911961624971455992, −10.64392297554471630400677505911, −9.980175470070455659993855969363, −9.069570123740443724390404064793, −8.105031387169368766400681276590, −7.14312272258317708388536898329, −6.34687988785922521290230471969, −4.85894487317697649364386311407, −3.25779803806112285509082430856, −1.63266017683856568882926995905, 1.63266017683856568882926995905, 3.25779803806112285509082430856, 4.85894487317697649364386311407, 6.34687988785922521290230471969, 7.14312272258317708388536898329, 8.105031387169368766400681276590, 9.069570123740443724390404064793, 9.980175470070455659993855969363, 10.64392297554471630400677505911, 11.66752597725911961624971455992

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