Properties

Label 1-328-328.245-r0-0-0
Degree $1$
Conductor $328$
Sign $1$
Analytic cond. $1.52322$
Root an. cond. $1.52322$
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  + 3-s − 5-s − 7-s + 9-s + 11-s + 13-s − 15-s − 17-s + 19-s − 21-s + 23-s + 25-s + 27-s + 29-s + 31-s + 33-s + 35-s − 37-s + 39-s − 43-s − 45-s − 47-s + 49-s − 51-s + 53-s − 55-s + 57-s + ⋯
L(s)  = 1  + 3-s − 5-s − 7-s + 9-s + 11-s + 13-s − 15-s − 17-s + 19-s − 21-s + 23-s + 25-s + 27-s + 29-s + 31-s + 33-s + 35-s − 37-s + 39-s − 43-s − 45-s − 47-s + 49-s − 51-s + 53-s − 55-s + 57-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(328\)    =    \(2^{3} \cdot 41\)
Sign: $1$
Analytic conductor: \(1.52322\)
Root analytic conductor: \(1.52322\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{328} (245, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 328,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.531918069\)
\(L(\frac12)\) \(\approx\) \(1.531918069\)
\(L(1)\) \(\approx\) \(1.278109739\)
\(L(1)\) \(\approx\) \(1.278109739\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−24.99045707055411689200971157309, −24.42572120051152116712836054226, −23.15115236523743386040375820338, −22.5546951894849845663600929362, −21.42068419239450792402641097942, −20.28905624371066057174354517029, −19.71256997854501670352616972201, −19.1001709563398781174376553789, −18.17179207667580366753056003303, −16.701145308605455284286213095356, −15.71891021599059229178484482279, −15.347239911669393001832921758103, −14.06699812599861350813215244403, −13.31590736200615575942602277263, −12.30946281954412819403769830274, −11.32110829513146335605344431515, −10.08227609551448378720438892784, −8.99734499009808474951469220330, −8.443279294674091008688208781748, −7.125493370537170429564279542072, −6.49468437690239370166774994232, −4.59912678691925579748294132327, −3.60899647730721711384272746565, −2.96703932597169844591223028468, −1.201938347406977831586729945566, 1.201938347406977831586729945566, 2.96703932597169844591223028468, 3.60899647730721711384272746565, 4.59912678691925579748294132327, 6.49468437690239370166774994232, 7.125493370537170429564279542072, 8.443279294674091008688208781748, 8.99734499009808474951469220330, 10.08227609551448378720438892784, 11.32110829513146335605344431515, 12.30946281954412819403769830274, 13.31590736200615575942602277263, 14.06699812599861350813215244403, 15.347239911669393001832921758103, 15.71891021599059229178484482279, 16.701145308605455284286213095356, 18.17179207667580366753056003303, 19.1001709563398781174376553789, 19.71256997854501670352616972201, 20.28905624371066057174354517029, 21.42068419239450792402641097942, 22.5546951894849845663600929362, 23.15115236523743386040375820338, 24.42572120051152116712836054226, 24.99045707055411689200971157309

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