Properties

Label 2-4788-1.1-c1-0-12
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  − 2.54·5-s − 7-s + 5.35·11-s + 6.80·13-s + 4.03·17-s − 19-s − 7.79·23-s + 1.45·25-s + 2.81·29-s − 1.54·31-s + 2.54·35-s + 5.09·37-s − 9.22·41-s + 1.54·43-s − 5.46·47-s + 49-s + 9.39·53-s − 13.6·55-s + 5.09·61-s − 17.2·65-s − 3.25·67-s + 1.32·71-s + 16.5·73-s − 5.35·77-s + 11.2·79-s − 6.19·83-s − 10.2·85-s + ⋯
L(s)  = 1  − 1.13·5-s − 0.377·7-s + 1.61·11-s + 1.88·13-s + 0.978·17-s − 0.229·19-s − 1.62·23-s + 0.290·25-s + 0.523·29-s − 0.277·31-s + 0.429·35-s + 0.837·37-s − 1.44·41-s + 0.235·43-s − 0.797·47-s + 0.142·49-s + 1.28·53-s − 1.83·55-s + 0.651·61-s − 2.14·65-s − 0.398·67-s + 0.157·71-s + 1.93·73-s − 0.610·77-s + 1.26·79-s − 0.679·83-s − 1.11·85-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

\(L(1)\) \(\approx\) \(1.731095414\)
\(L(\frac12)\) \(\approx\) \(1.731095414\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.54T + 5T^{2} \)
11 \( 1 - 5.35T + 11T^{2} \)
13 \( 1 - 6.80T + 13T^{2} \)
17 \( 1 - 4.03T + 17T^{2} \)
23 \( 1 + 7.79T + 23T^{2} \)
29 \( 1 - 2.81T + 29T^{2} \)
31 \( 1 + 1.54T + 31T^{2} \)
37 \( 1 - 5.09T + 37T^{2} \)
41 \( 1 + 9.22T + 41T^{2} \)
43 \( 1 - 1.54T + 43T^{2} \)
47 \( 1 + 5.46T + 47T^{2} \)
53 \( 1 - 9.39T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 5.09T + 61T^{2} \)
67 \( 1 + 3.25T + 67T^{2} \)
71 \( 1 - 1.32T + 71T^{2} \)
73 \( 1 - 16.5T + 73T^{2} \)
79 \( 1 - 11.2T + 79T^{2} \)
83 \( 1 + 6.19T + 83T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 17.9T + 97T^{2} \)
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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

−8.299188409379086985163201407174, −7.70978490289662335078219414827, −6.66309533391090313085671419640, −6.31409246288072125601485836429, −5.46708088910072345842167089718, −4.08050975262303829175866257678, −3.91820568119709272520041352285, −3.23757888162928826721489264790, −1.71681658209574510326074413655, −0.76654623292882484452203057510, 0.76654623292882484452203057510, 1.71681658209574510326074413655, 3.23757888162928826721489264790, 3.91820568119709272520041352285, 4.08050975262303829175866257678, 5.46708088910072345842167089718, 6.31409246288072125601485836429, 6.66309533391090313085671419640, 7.70978490289662335078219414827, 8.299188409379086985163201407174

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