Properties

Label 2-4864-1.1-c1-0-117
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  − 0.458·3-s + 3.00·5-s + 2.21·7-s − 2.79·9-s − 5.02·11-s − 5.22·13-s − 1.37·15-s + 7.73·17-s − 19-s − 1.01·21-s − 6.51·23-s + 4.02·25-s + 2.65·27-s + 2.23·29-s − 3.33·31-s + 2.30·33-s + 6.65·35-s + 1.45·37-s + 2.39·39-s + 3.16·41-s + 2.16·43-s − 8.38·45-s + 9.95·47-s − 2.08·49-s − 3.54·51-s − 6.66·53-s − 15.0·55-s + ⋯
L(s)  = 1  − 0.264·3-s + 1.34·5-s + 0.837·7-s − 0.930·9-s − 1.51·11-s − 1.44·13-s − 0.355·15-s + 1.87·17-s − 0.229·19-s − 0.221·21-s − 1.35·23-s + 0.804·25-s + 0.510·27-s + 0.415·29-s − 0.599·31-s + 0.400·33-s + 1.12·35-s + 0.238·37-s + 0.382·39-s + 0.494·41-s + 0.329·43-s − 1.24·45-s + 1.45·47-s − 0.298·49-s − 0.495·51-s − 0.915·53-s − 2.03·55-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 \( 1 + 0.458T + 3T^{2} \)
5 \( 1 - 3.00T + 5T^{2} \)
7 \( 1 - 2.21T + 7T^{2} \)
11 \( 1 + 5.02T + 11T^{2} \)
13 \( 1 + 5.22T + 13T^{2} \)
17 \( 1 - 7.73T + 17T^{2} \)
23 \( 1 + 6.51T + 23T^{2} \)
29 \( 1 - 2.23T + 29T^{2} \)
31 \( 1 + 3.33T + 31T^{2} \)
37 \( 1 - 1.45T + 37T^{2} \)
41 \( 1 - 3.16T + 41T^{2} \)
43 \( 1 - 2.16T + 43T^{2} \)
47 \( 1 - 9.95T + 47T^{2} \)
53 \( 1 + 6.66T + 53T^{2} \)
59 \( 1 - 3.89T + 59T^{2} \)
61 \( 1 + 7.05T + 61T^{2} \)
67 \( 1 + 13.2T + 67T^{2} \)
71 \( 1 + 5.72T + 71T^{2} \)
73 \( 1 + 5.89T + 73T^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 + 8.71T + 83T^{2} \)
89 \( 1 + 3.79T + 89T^{2} \)
97 \( 1 + 3.94T + 97T^{2} \)
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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

−7.74039113938318155665684957337, −7.48869216416204053383042446562, −6.11502726753987223164932113221, −5.59750427330962030756312228407, −5.27714879181561139342743033409, −4.43449400425858611395017677702, −2.92803713786583605740375977516, −2.46294670831354902471378202681, −1.51870112462374108445921288617, 0, 1.51870112462374108445921288617, 2.46294670831354902471378202681, 2.92803713786583605740375977516, 4.43449400425858611395017677702, 5.27714879181561139342743033409, 5.59750427330962030756312228407, 6.11502726753987223164932113221, 7.48869216416204053383042446562, 7.74039113938318155665684957337

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