Properties

Label 2-4864-1.1-c1-0-138
Degree $2$
Conductor $4864$
Sign $-1$
Analytic cond. $38.8392$
Root an. cond. $6.23211$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.91·3-s + 1.51·5-s − 0.580·7-s + 0.682·9-s − 1.31·11-s − 3.89·13-s + 2.90·15-s − 1.20·17-s − 19-s − 1.11·21-s − 5.85·23-s − 2.70·25-s − 4.44·27-s − 1.29·29-s + 2.96·31-s − 2.52·33-s − 0.878·35-s − 1.18·37-s − 7.46·39-s + 9.04·41-s + 8.38·43-s + 1.03·45-s − 12.8·47-s − 6.66·49-s − 2.30·51-s − 3.07·53-s − 1.99·55-s + ⋯
L(s)  = 1  + 1.10·3-s + 0.676·5-s − 0.219·7-s + 0.227·9-s − 0.397·11-s − 1.07·13-s + 0.749·15-s − 0.291·17-s − 0.229·19-s − 0.242·21-s − 1.22·23-s − 0.541·25-s − 0.855·27-s − 0.239·29-s + 0.532·31-s − 0.440·33-s − 0.148·35-s − 0.194·37-s − 1.19·39-s + 1.41·41-s + 1.27·43-s + 0.153·45-s − 1.87·47-s − 0.951·49-s − 0.322·51-s − 0.421·53-s − 0.268·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4864\)    =    \(2^{8} \cdot 19\)
Sign: $-1$
Analytic conductor: \(38.8392\)
Root analytic conductor: \(6.23211\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4864} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4864,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + T \)
good3 \( 1 - 1.91T + 3T^{2} \)
5 \( 1 - 1.51T + 5T^{2} \)
7 \( 1 + 0.580T + 7T^{2} \)
11 \( 1 + 1.31T + 11T^{2} \)
13 \( 1 + 3.89T + 13T^{2} \)
17 \( 1 + 1.20T + 17T^{2} \)
23 \( 1 + 5.85T + 23T^{2} \)
29 \( 1 + 1.29T + 29T^{2} \)
31 \( 1 - 2.96T + 31T^{2} \)
37 \( 1 + 1.18T + 37T^{2} \)
41 \( 1 - 9.04T + 41T^{2} \)
43 \( 1 - 8.38T + 43T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 + 3.07T + 53T^{2} \)
59 \( 1 - 0.258T + 59T^{2} \)
61 \( 1 + 14.7T + 61T^{2} \)
67 \( 1 - 9.54T + 67T^{2} \)
71 \( 1 - 6.93T + 71T^{2} \)
73 \( 1 + 15.2T + 73T^{2} \)
79 \( 1 + 13.1T + 79T^{2} \)
83 \( 1 - 3.70T + 83T^{2} \)
89 \( 1 - 8.36T + 89T^{2} \)
97 \( 1 - 17.0T + 97T^{2} \)
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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

−7.84537847176014262226180922781, −7.51356241765214218527406472983, −6.37625740446703437091975606014, −5.84748794945134859779076247465, −4.87997659696316473972561144716, −4.07816597778407493225569689611, −3.09910138546274862994982537276, −2.42176674705161307707437156601, −1.79347758528340037338841492850, 0, 1.79347758528340037338841492850, 2.42176674705161307707437156601, 3.09910138546274862994982537276, 4.07816597778407493225569689611, 4.87997659696316473972561144716, 5.84748794945134859779076247465, 6.37625740446703437091975606014, 7.51356241765214218527406472983, 7.84537847176014262226180922781

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