Properties

Label 2-4864-1.1-c1-0-127
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  + 2.95·3-s − 2.13·5-s − 3.29·7-s + 5.71·9-s + 3.71·11-s + 2.32·13-s − 6.30·15-s − 6.48·17-s − 19-s − 9.72·21-s − 7.32·23-s − 0.442·25-s + 8.00·27-s − 2.59·29-s − 1.34·31-s + 10.9·33-s + 7.03·35-s + 3.72·37-s + 6.87·39-s − 6.52·41-s + 1.97·43-s − 12.1·45-s + 5.45·47-s + 3.85·49-s − 19.1·51-s − 4.98·53-s − 7.92·55-s + ⋯
L(s)  = 1  + 1.70·3-s − 0.954·5-s − 1.24·7-s + 1.90·9-s + 1.11·11-s + 0.645·13-s − 1.62·15-s − 1.57·17-s − 0.229·19-s − 2.12·21-s − 1.52·23-s − 0.0884·25-s + 1.54·27-s − 0.482·29-s − 0.241·31-s + 1.90·33-s + 1.18·35-s + 0.613·37-s + 1.10·39-s − 1.01·41-s + 0.300·43-s − 1.81·45-s + 0.796·47-s + 0.550·49-s − 2.68·51-s − 0.684·53-s − 1.06·55-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 \( 1 - 2.95T + 3T^{2} \)
5 \( 1 + 2.13T + 5T^{2} \)
7 \( 1 + 3.29T + 7T^{2} \)
11 \( 1 - 3.71T + 11T^{2} \)
13 \( 1 - 2.32T + 13T^{2} \)
17 \( 1 + 6.48T + 17T^{2} \)
23 \( 1 + 7.32T + 23T^{2} \)
29 \( 1 + 2.59T + 29T^{2} \)
31 \( 1 + 1.34T + 31T^{2} \)
37 \( 1 - 3.72T + 37T^{2} \)
41 \( 1 + 6.52T + 41T^{2} \)
43 \( 1 - 1.97T + 43T^{2} \)
47 \( 1 - 5.45T + 47T^{2} \)
53 \( 1 + 4.98T + 53T^{2} \)
59 \( 1 + 9.67T + 59T^{2} \)
61 \( 1 - 8.15T + 61T^{2} \)
67 \( 1 + 0.524T + 67T^{2} \)
71 \( 1 + 7.17T + 71T^{2} \)
73 \( 1 - 6.33T + 73T^{2} \)
79 \( 1 + 8.75T + 79T^{2} \)
83 \( 1 + 7.74T + 83T^{2} \)
89 \( 1 - 1.04T + 89T^{2} \)
97 \( 1 + 0.117T + 97T^{2} \)
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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

−8.072380285126371522226714205043, −7.29952135290414194470340237993, −6.64137197106691853394082144613, −6.03537637064964074128067302771, −4.35599733197355413834455492887, −3.93540559574984392906441192802, −3.47079258820108530601539696782, −2.56044233728173170470161715899, −1.65482518631527544622499545872, 0, 1.65482518631527544622499545872, 2.56044233728173170470161715899, 3.47079258820108530601539696782, 3.93540559574984392906441192802, 4.35599733197355413834455492887, 6.03537637064964074128067302771, 6.64137197106691853394082144613, 7.29952135290414194470340237993, 8.072380285126371522226714205043

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