Properties

Label 2-4024-1.1-c1-0-83
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  + 2.03·3-s + 4.46·5-s + 0.828·7-s + 1.13·9-s + 0.0966·11-s + 1.53·13-s + 9.07·15-s − 3.59·17-s + 8.33·19-s + 1.68·21-s + 2.93·23-s + 14.9·25-s − 3.79·27-s + 1.38·29-s − 2.48·31-s + 0.196·33-s + 3.69·35-s − 3.06·37-s + 3.12·39-s − 5.09·41-s − 12.0·43-s + 5.05·45-s − 3.75·47-s − 6.31·49-s − 7.30·51-s + 5.98·53-s + 0.431·55-s + ⋯
L(s)  = 1  + 1.17·3-s + 1.99·5-s + 0.313·7-s + 0.378·9-s + 0.0291·11-s + 0.426·13-s + 2.34·15-s − 0.871·17-s + 1.91·19-s + 0.367·21-s + 0.611·23-s + 2.98·25-s − 0.730·27-s + 0.256·29-s − 0.446·31-s + 0.0342·33-s + 0.625·35-s − 0.504·37-s + 0.501·39-s − 0.794·41-s − 1.84·43-s + 0.754·45-s − 0.547·47-s − 0.901·49-s − 1.02·51-s + 0.821·53-s + 0.0581·55-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

\(L(1)\) \(\approx\) \(4.647709072\)
\(L(\frac12)\) \(\approx\) \(4.647709072\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.03T + 3T^{2} \)
5 \( 1 - 4.46T + 5T^{2} \)
7 \( 1 - 0.828T + 7T^{2} \)
11 \( 1 - 0.0966T + 11T^{2} \)
13 \( 1 - 1.53T + 13T^{2} \)
17 \( 1 + 3.59T + 17T^{2} \)
19 \( 1 - 8.33T + 19T^{2} \)
23 \( 1 - 2.93T + 23T^{2} \)
29 \( 1 - 1.38T + 29T^{2} \)
31 \( 1 + 2.48T + 31T^{2} \)
37 \( 1 + 3.06T + 37T^{2} \)
41 \( 1 + 5.09T + 41T^{2} \)
43 \( 1 + 12.0T + 43T^{2} \)
47 \( 1 + 3.75T + 47T^{2} \)
53 \( 1 - 5.98T + 53T^{2} \)
59 \( 1 - 1.55T + 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 - 1.19T + 67T^{2} \)
71 \( 1 + 5.48T + 71T^{2} \)
73 \( 1 - 5.93T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 16.7T + 83T^{2} \)
89 \( 1 + 7.41T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−8.695513591330745483807597509280, −7.85561908890381062990494791237, −6.90991341563468220833483682310, −6.31567046547131633354882786576, −5.33064407510800716926984271634, −4.95120284214288698801550515334, −3.47409875320108755735511587672, −2.88102092686842330680405565995, −1.98601260845425265713941480362, −1.36108621116723172056489891470, 1.36108621116723172056489891470, 1.98601260845425265713941480362, 2.88102092686842330680405565995, 3.47409875320108755735511587672, 4.95120284214288698801550515334, 5.33064407510800716926984271634, 6.31567046547131633354882786576, 6.90991341563468220833483682310, 7.85561908890381062990494791237, 8.695513591330745483807597509280

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