Properties

Label 2-4788-1.1-c1-0-16
Degree $2$
Conductor $4788$
Sign $1$
Analytic cond. $38.2323$
Root an. cond. $6.18323$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 3.46·11-s − 4·13-s + 6.92·17-s + 19-s + 3.46·23-s − 5·25-s − 3.46·29-s + 8·31-s + 2·37-s − 6.92·41-s + 8·43-s + 3.46·47-s + 49-s − 10.3·53-s − 13.8·59-s + 2·61-s + 2·67-s + 3.46·71-s + 14·73-s + 3.46·77-s + 14·79-s − 10.3·83-s + 13.8·89-s − 4·91-s + 8·97-s − 4·103-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.04·11-s − 1.10·13-s + 1.68·17-s + 0.229·19-s + 0.722·23-s − 25-s − 0.643·29-s + 1.43·31-s + 0.328·37-s − 1.08·41-s + 1.21·43-s + 0.505·47-s + 0.142·49-s − 1.42·53-s − 1.80·59-s + 0.256·61-s + 0.244·67-s + 0.411·71-s + 1.63·73-s + 0.394·77-s + 1.57·79-s − 1.14·83-s + 1.46·89-s − 0.419·91-s + 0.812·97-s − 0.394·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4788 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4788\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(38.2323\)
Root analytic conductor: \(6.18323\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4788,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.224867115\)
\(L(\frac12)\) \(\approx\) \(2.224867115\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - T \)
19 \( 1 - T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 6.92T + 17T^{2} \)
23 \( 1 - 3.46T + 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 6.92T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 13.8T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 - 2T + 67T^{2} \)
71 \( 1 - 3.46T + 71T^{2} \)
73 \( 1 - 14T + 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−7.996994774262594005284166465632, −7.75073681995543965464618590410, −6.87037150575240516729161690830, −6.10860374961603052350653433191, −5.30678880024795367109817291558, −4.63328745194643210287311083983, −3.73239630914176293392467645183, −2.93600523775511834788550324634, −1.83887735495383727690659992435, −0.859489601820127144261081688810, 0.859489601820127144261081688810, 1.83887735495383727690659992435, 2.93600523775511834788550324634, 3.73239630914176293392467645183, 4.63328745194643210287311083983, 5.30678880024795367109817291558, 6.10860374961603052350653433191, 6.87037150575240516729161690830, 7.75073681995543965464618590410, 7.996994774262594005284166465632

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