Properties

Label 2-4864-1.1-c1-0-109
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  + 0.222·3-s + 1.55·5-s − 3.06·7-s − 2.95·9-s + 0.950·11-s + 3.36·13-s + 0.344·15-s + 1.82·17-s + 19-s − 0.680·21-s − 3.36·23-s − 2.59·25-s − 1.32·27-s + 4.95·29-s − 3.44·31-s + 0.211·33-s − 4.75·35-s − 2.95·37-s + 0.746·39-s − 4.55·41-s − 1.69·43-s − 4.57·45-s − 3.39·47-s + 2.38·49-s + 0.406·51-s + 4.47·53-s + 1.47·55-s + ⋯
L(s)  = 1  + 0.128·3-s + 0.693·5-s − 1.15·7-s − 0.983·9-s + 0.286·11-s + 0.932·13-s + 0.0889·15-s + 0.443·17-s + 0.229·19-s − 0.148·21-s − 0.700·23-s − 0.518·25-s − 0.254·27-s + 0.920·29-s − 0.618·31-s + 0.0367·33-s − 0.803·35-s − 0.486·37-s + 0.119·39-s − 0.711·41-s − 0.258·43-s − 0.682·45-s − 0.495·47-s + 0.340·49-s + 0.0568·51-s + 0.614·53-s + 0.198·55-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 \( 1 - 0.222T + 3T^{2} \)
5 \( 1 - 1.55T + 5T^{2} \)
7 \( 1 + 3.06T + 7T^{2} \)
11 \( 1 - 0.950T + 11T^{2} \)
13 \( 1 - 3.36T + 13T^{2} \)
17 \( 1 - 1.82T + 17T^{2} \)
23 \( 1 + 3.36T + 23T^{2} \)
29 \( 1 - 4.95T + 29T^{2} \)
31 \( 1 + 3.44T + 31T^{2} \)
37 \( 1 + 2.95T + 37T^{2} \)
41 \( 1 + 4.55T + 41T^{2} \)
43 \( 1 + 1.69T + 43T^{2} \)
47 \( 1 + 3.39T + 47T^{2} \)
53 \( 1 - 4.47T + 53T^{2} \)
59 \( 1 - 0.395T + 59T^{2} \)
61 \( 1 - 5.34T + 61T^{2} \)
67 \( 1 + 6.85T + 67T^{2} \)
71 \( 1 - 5.15T + 71T^{2} \)
73 \( 1 + 1.19T + 73T^{2} \)
79 \( 1 + 14.7T + 79T^{2} \)
83 \( 1 + 3.35T + 83T^{2} \)
89 \( 1 + 9.64T + 89T^{2} \)
97 \( 1 - 3.55T + 97T^{2} \)
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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

−8.065462850331454225433900156657, −7.03733911796613430770222664344, −6.29253080515091415049547869624, −5.89396066696909245377713367409, −5.18505744620279523188507464202, −3.90126185235079611782421943246, −3.31598030643141890543502511638, −2.51098769073577638311793478918, −1.41556585882987888645439229434, 0, 1.41556585882987888645439229434, 2.51098769073577638311793478918, 3.31598030643141890543502511638, 3.90126185235079611782421943246, 5.18505744620279523188507464202, 5.89396066696909245377713367409, 6.29253080515091415049547869624, 7.03733911796613430770222664344, 8.065462850331454225433900156657

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