Properties

Label 2-384-1.1-c3-0-1
Degree $2$
Conductor $384$
Sign $1$
Analytic cond. $22.6567$
Root an. cond. $4.75990$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 19.4·5-s + 17.4·7-s + 9·9-s − 50.9·11-s − 2·13-s + 58.4·15-s − 78.9·17-s − 25.0·19-s − 52.4·21-s − 120.·23-s + 254.·25-s − 27·27-s + 273.·29-s + 29.5·31-s + 152.·33-s − 340.·35-s + 320.·37-s + 6·39-s + 263.·41-s + 388.·43-s − 175.·45-s + 325.·47-s − 37.0·49-s + 236.·51-s + 567.·53-s + 993.·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.74·5-s + 0.944·7-s + 0.333·9-s − 1.39·11-s − 0.0426·13-s + 1.00·15-s − 1.12·17-s − 0.302·19-s − 0.545·21-s − 1.09·23-s + 2.03·25-s − 0.192·27-s + 1.75·29-s + 0.171·31-s + 0.806·33-s − 1.64·35-s + 1.42·37-s + 0.0246·39-s + 1.00·41-s + 1.37·43-s − 0.581·45-s + 1.00·47-s − 0.107·49-s + 0.650·51-s + 1.47·53-s + 2.43·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $1$
Analytic conductor: \(22.6567\)
Root analytic conductor: \(4.75990\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 384,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7852871268\)
\(L(\frac12)\) \(\approx\) \(0.7852871268\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + 3T \)
good5 \( 1 + 19.4T + 125T^{2} \)
7 \( 1 - 17.4T + 343T^{2} \)
11 \( 1 + 50.9T + 1.33e3T^{2} \)
13 \( 1 + 2T + 2.19e3T^{2} \)
17 \( 1 + 78.9T + 4.91e3T^{2} \)
19 \( 1 + 25.0T + 6.85e3T^{2} \)
23 \( 1 + 120.T + 1.21e4T^{2} \)
29 \( 1 - 273.T + 2.43e4T^{2} \)
31 \( 1 - 29.5T + 2.97e4T^{2} \)
37 \( 1 - 320.T + 5.06e4T^{2} \)
41 \( 1 - 263.T + 6.89e4T^{2} \)
43 \( 1 - 388.T + 7.95e4T^{2} \)
47 \( 1 - 325.T + 1.03e5T^{2} \)
53 \( 1 - 567.T + 1.48e5T^{2} \)
59 \( 1 + 639.T + 2.05e5T^{2} \)
61 \( 1 + 176.T + 2.26e5T^{2} \)
67 \( 1 + 1.01e3T + 3.00e5T^{2} \)
71 \( 1 - 15.0T + 3.57e5T^{2} \)
73 \( 1 + 623.T + 3.89e5T^{2} \)
79 \( 1 - 305.T + 4.93e5T^{2} \)
83 \( 1 - 636.T + 5.71e5T^{2} \)
89 \( 1 - 1.46e3T + 7.04e5T^{2} \)
97 \( 1 + 640.T + 9.12e5T^{2} \)
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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

−10.93312570741235954135239994276, −10.48318308232895671196274876193, −8.801812101308625975588884186281, −7.899461321787048714281473271949, −7.48545142279542202445989797800, −6.09818303186692467824397556239, −4.69552828929516958014933929651, −4.27440332344970367894261646307, −2.58549956106061615759176628569, −0.58296570279222121841720299447, 0.58296570279222121841720299447, 2.58549956106061615759176628569, 4.27440332344970367894261646307, 4.69552828929516958014933929651, 6.09818303186692467824397556239, 7.48545142279542202445989797800, 7.899461321787048714281473271949, 8.801812101308625975588884186281, 10.48318308232895671196274876193, 10.93312570741235954135239994276

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