Properties

Label 2-4024-1.1-c1-0-92
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  + 2.66·3-s + 3.83·5-s + 0.592·7-s + 4.08·9-s + 2.69·11-s − 0.914·13-s + 10.1·15-s + 4.61·17-s − 4.04·19-s + 1.57·21-s + 1.00·23-s + 9.66·25-s + 2.87·27-s − 0.102·29-s − 5.16·31-s + 7.16·33-s + 2.26·35-s + 1.33·37-s − 2.43·39-s − 8.01·41-s − 0.552·43-s + 15.6·45-s + 4.45·47-s − 6.64·49-s + 12.2·51-s − 4.69·53-s + 10.3·55-s + ⋯
L(s)  = 1  + 1.53·3-s + 1.71·5-s + 0.223·7-s + 1.36·9-s + 0.811·11-s − 0.253·13-s + 2.63·15-s + 1.11·17-s − 0.928·19-s + 0.344·21-s + 0.209·23-s + 1.93·25-s + 0.553·27-s − 0.0190·29-s − 0.926·31-s + 1.24·33-s + 0.383·35-s + 0.219·37-s − 0.389·39-s − 1.25·41-s − 0.0842·43-s + 2.33·45-s + 0.650·47-s − 0.949·49-s + 1.72·51-s − 0.644·53-s + 1.39·55-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

\(L(1)\) \(\approx\) \(5.103340828\)
\(L(\frac12)\) \(\approx\) \(5.103340828\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.66T + 3T^{2} \)
5 \( 1 - 3.83T + 5T^{2} \)
7 \( 1 - 0.592T + 7T^{2} \)
11 \( 1 - 2.69T + 11T^{2} \)
13 \( 1 + 0.914T + 13T^{2} \)
17 \( 1 - 4.61T + 17T^{2} \)
19 \( 1 + 4.04T + 19T^{2} \)
23 \( 1 - 1.00T + 23T^{2} \)
29 \( 1 + 0.102T + 29T^{2} \)
31 \( 1 + 5.16T + 31T^{2} \)
37 \( 1 - 1.33T + 37T^{2} \)
41 \( 1 + 8.01T + 41T^{2} \)
43 \( 1 + 0.552T + 43T^{2} \)
47 \( 1 - 4.45T + 47T^{2} \)
53 \( 1 + 4.69T + 53T^{2} \)
59 \( 1 + 7.37T + 59T^{2} \)
61 \( 1 - 5.46T + 61T^{2} \)
67 \( 1 + 8.16T + 67T^{2} \)
71 \( 1 + 7.48T + 71T^{2} \)
73 \( 1 - 7.15T + 73T^{2} \)
79 \( 1 - 1.67T + 79T^{2} \)
83 \( 1 + 1.36T + 83T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 - 11.9T + 97T^{2} \)
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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

−8.647129967616994041952287548741, −7.85417200217601678962115046760, −7.03500619439244947997337684593, −6.27763540088032400803751510169, −5.52593167094951851168426286613, −4.61890171898097387694681658619, −3.58837342715775426153090758619, −2.85192991759583626566753682028, −1.95963746535171816104512498311, −1.45492879689212154191929150668, 1.45492879689212154191929150668, 1.95963746535171816104512498311, 2.85192991759583626566753682028, 3.58837342715775426153090758619, 4.61890171898097387694681658619, 5.52593167094951851168426286613, 6.27763540088032400803751510169, 7.03500619439244947997337684593, 7.85417200217601678962115046760, 8.647129967616994041952287548741

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