Properties

Label 2-4024-1.1-c1-0-19
Degree $2$
Conductor $4024$
Sign $1$
Analytic cond. $32.1318$
Root an. cond. $5.66849$
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  − 2.71·3-s − 1.51·5-s − 2.20·7-s + 4.35·9-s + 6.02·11-s + 4.45·13-s + 4.11·15-s − 3.55·17-s + 3.54·19-s + 5.98·21-s − 4.23·23-s − 2.69·25-s − 3.66·27-s + 1.06·29-s + 6.50·31-s − 16.3·33-s + 3.35·35-s − 10.6·37-s − 12.0·39-s + 6.73·41-s + 10.4·43-s − 6.60·45-s + 5.48·47-s − 2.13·49-s + 9.63·51-s − 5.47·53-s − 9.15·55-s + ⋯
L(s)  = 1  − 1.56·3-s − 0.679·5-s − 0.834·7-s + 1.45·9-s + 1.81·11-s + 1.23·13-s + 1.06·15-s − 0.862·17-s + 0.814·19-s + 1.30·21-s − 0.883·23-s − 0.538·25-s − 0.705·27-s + 0.198·29-s + 1.16·31-s − 2.84·33-s + 0.566·35-s − 1.75·37-s − 1.93·39-s + 1.05·41-s + 1.59·43-s − 0.985·45-s + 0.800·47-s − 0.304·49-s + 1.34·51-s − 0.751·53-s − 1.23·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $1$
Analytic conductor: \(32.1318\)
Root analytic conductor: \(5.66849\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4024,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8088951802\)
\(L(\frac12)\) \(\approx\) \(0.8088951802\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 - T \)
good3 \( 1 + 2.71T + 3T^{2} \)
5 \( 1 + 1.51T + 5T^{2} \)
7 \( 1 + 2.20T + 7T^{2} \)
11 \( 1 - 6.02T + 11T^{2} \)
13 \( 1 - 4.45T + 13T^{2} \)
17 \( 1 + 3.55T + 17T^{2} \)
19 \( 1 - 3.54T + 19T^{2} \)
23 \( 1 + 4.23T + 23T^{2} \)
29 \( 1 - 1.06T + 29T^{2} \)
31 \( 1 - 6.50T + 31T^{2} \)
37 \( 1 + 10.6T + 37T^{2} \)
41 \( 1 - 6.73T + 41T^{2} \)
43 \( 1 - 10.4T + 43T^{2} \)
47 \( 1 - 5.48T + 47T^{2} \)
53 \( 1 + 5.47T + 53T^{2} \)
59 \( 1 + 9.11T + 59T^{2} \)
61 \( 1 + 5.04T + 61T^{2} \)
67 \( 1 + 0.260T + 67T^{2} \)
71 \( 1 - 14.2T + 71T^{2} \)
73 \( 1 + 15.3T + 73T^{2} \)
79 \( 1 + 12.2T + 79T^{2} \)
83 \( 1 + 16.8T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 - 8.93T + 97T^{2} \)
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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

−8.540879674525277755332443939945, −7.43237431766229211686526817612, −6.75569092647686841948189375328, −6.10683611704082054404270864213, −5.87154213648557728677039874198, −4.50859991625115224744828724997, −4.07424541096297084129476960016, −3.25434140374442194904690213930, −1.54014425857508689926187043472, −0.59550063941962858215518312657, 0.59550063941962858215518312657, 1.54014425857508689926187043472, 3.25434140374442194904690213930, 4.07424541096297084129476960016, 4.50859991625115224744828724997, 5.87154213648557728677039874198, 6.10683611704082054404270864213, 6.75569092647686841948189375328, 7.43237431766229211686526817612, 8.540879674525277755332443939945

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