Properties

Label 2-4024-1.1-c1-0-55
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  − 3.01·3-s + 3.98·5-s + 4.01·7-s + 6.07·9-s − 1.62·11-s + 0.371·13-s − 12.0·15-s + 4.12·17-s + 6.23·19-s − 12.1·21-s − 1.25·23-s + 10.9·25-s − 9.25·27-s + 4.56·29-s − 3.83·31-s + 4.89·33-s + 16.0·35-s + 5.64·37-s − 1.11·39-s − 8.40·41-s + 0.628·43-s + 24.2·45-s + 10.0·47-s + 9.14·49-s − 12.4·51-s + 7.64·53-s − 6.48·55-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.78·5-s + 1.51·7-s + 2.02·9-s − 0.490·11-s + 0.102·13-s − 3.10·15-s + 1.00·17-s + 1.42·19-s − 2.64·21-s − 0.260·23-s + 2.18·25-s − 1.78·27-s + 0.847·29-s − 0.688·31-s + 0.852·33-s + 2.70·35-s + 0.928·37-s − 0.178·39-s − 1.31·41-s + 0.0958·43-s + 3.60·45-s + 1.46·47-s + 1.30·49-s − 1.74·51-s + 1.04·53-s − 0.874·55-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

\(L(1)\) \(\approx\) \(2.095944703\)
\(L(\frac12)\) \(\approx\) \(2.095944703\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.01T + 3T^{2} \)
5 \( 1 - 3.98T + 5T^{2} \)
7 \( 1 - 4.01T + 7T^{2} \)
11 \( 1 + 1.62T + 11T^{2} \)
13 \( 1 - 0.371T + 13T^{2} \)
17 \( 1 - 4.12T + 17T^{2} \)
19 \( 1 - 6.23T + 19T^{2} \)
23 \( 1 + 1.25T + 23T^{2} \)
29 \( 1 - 4.56T + 29T^{2} \)
31 \( 1 + 3.83T + 31T^{2} \)
37 \( 1 - 5.64T + 37T^{2} \)
41 \( 1 + 8.40T + 41T^{2} \)
43 \( 1 - 0.628T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 - 7.64T + 53T^{2} \)
59 \( 1 - 2.62T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + 7.23T + 67T^{2} \)
71 \( 1 + 9.50T + 71T^{2} \)
73 \( 1 + 10.2T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 - 2.15T + 83T^{2} \)
89 \( 1 - 8.00T + 89T^{2} \)
97 \( 1 + 11.6T + 97T^{2} \)
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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

−8.416125101765493409482568863931, −7.48785892119988203065438771311, −6.84470383007059697801257833912, −5.84739327455916319132084794240, −5.45236849147856397947755813050, −5.18940237364768255519031591046, −4.27414172174715154584473574152, −2.68208390529119643345358392003, −1.54612308837083997841025302563, −1.05611100049065067351691676016, 1.05611100049065067351691676016, 1.54612308837083997841025302563, 2.68208390529119643345358392003, 4.27414172174715154584473574152, 5.18940237364768255519031591046, 5.45236849147856397947755813050, 5.84739327455916319132084794240, 6.84470383007059697801257833912, 7.48785892119988203065438771311, 8.416125101765493409482568863931

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