Properties

Label 2-4024-1.1-c1-0-74
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  + 1.78·3-s − 0.988·5-s + 3.73·7-s + 0.196·9-s + 5.76·11-s + 4.01·13-s − 1.76·15-s + 6.16·17-s + 4.14·19-s + 6.68·21-s − 1.16·23-s − 4.02·25-s − 5.01·27-s + 5.75·29-s + 1.16·31-s + 10.3·33-s − 3.69·35-s − 9.17·37-s + 7.18·39-s − 7.67·41-s − 3.71·43-s − 0.194·45-s − 5.48·47-s + 6.96·49-s + 11.0·51-s + 1.03·53-s − 5.69·55-s + ⋯
L(s)  = 1  + 1.03·3-s − 0.441·5-s + 1.41·7-s + 0.0656·9-s + 1.73·11-s + 1.11·13-s − 0.456·15-s + 1.49·17-s + 0.950·19-s + 1.45·21-s − 0.243·23-s − 0.804·25-s − 0.964·27-s + 1.06·29-s + 0.209·31-s + 1.79·33-s − 0.624·35-s − 1.50·37-s + 1.14·39-s − 1.19·41-s − 0.566·43-s − 0.0290·45-s − 0.800·47-s + 0.995·49-s + 1.54·51-s + 0.141·53-s − 0.767·55-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

\(L(1)\) \(\approx\) \(3.725848196\)
\(L(\frac12)\) \(\approx\) \(3.725848196\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.78T + 3T^{2} \)
5 \( 1 + 0.988T + 5T^{2} \)
7 \( 1 - 3.73T + 7T^{2} \)
11 \( 1 - 5.76T + 11T^{2} \)
13 \( 1 - 4.01T + 13T^{2} \)
17 \( 1 - 6.16T + 17T^{2} \)
19 \( 1 - 4.14T + 19T^{2} \)
23 \( 1 + 1.16T + 23T^{2} \)
29 \( 1 - 5.75T + 29T^{2} \)
31 \( 1 - 1.16T + 31T^{2} \)
37 \( 1 + 9.17T + 37T^{2} \)
41 \( 1 + 7.67T + 41T^{2} \)
43 \( 1 + 3.71T + 43T^{2} \)
47 \( 1 + 5.48T + 47T^{2} \)
53 \( 1 - 1.03T + 53T^{2} \)
59 \( 1 + 5.95T + 59T^{2} \)
61 \( 1 + 2.55T + 61T^{2} \)
67 \( 1 + 9.37T + 67T^{2} \)
71 \( 1 + 4.55T + 71T^{2} \)
73 \( 1 - 11.9T + 73T^{2} \)
79 \( 1 + 9.39T + 79T^{2} \)
83 \( 1 - 3.37T + 83T^{2} \)
89 \( 1 + 0.772T + 89T^{2} \)
97 \( 1 - 16.9T + 97T^{2} \)
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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

−8.367679278363138151289081983142, −7.971731182174413136950225038385, −7.22622426831850005252584115165, −6.26889976354317132710524012510, −5.41839386673032097924420447314, −4.50666787099988380365813034114, −3.57995769002558753268406484666, −3.29077325305257511130046002703, −1.74426970449373453527184949709, −1.25387526133940368292327720643, 1.25387526133940368292327720643, 1.74426970449373453527184949709, 3.29077325305257511130046002703, 3.57995769002558753268406484666, 4.50666787099988380365813034114, 5.41839386673032097924420447314, 6.26889976354317132710524012510, 7.22622426831850005252584115165, 7.971731182174413136950225038385, 8.367679278363138151289081983142

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