Properties

Label 2-4024-1.1-c1-0-39
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  + 2.60·3-s − 1.79·5-s − 2.96·7-s + 3.80·9-s − 0.556·11-s + 3.76·13-s − 4.69·15-s + 1.41·17-s + 1.17·19-s − 7.72·21-s + 0.626·23-s − 1.76·25-s + 2.11·27-s − 0.654·29-s + 1.67·31-s − 1.45·33-s + 5.32·35-s − 2.59·37-s + 9.83·39-s + 6.68·41-s + 11.7·43-s − 6.85·45-s + 13.0·47-s + 1.76·49-s + 3.69·51-s + 9.66·53-s + 1.00·55-s + ⋯
L(s)  = 1  + 1.50·3-s − 0.804·5-s − 1.11·7-s + 1.26·9-s − 0.167·11-s + 1.04·13-s − 1.21·15-s + 0.343·17-s + 0.270·19-s − 1.68·21-s + 0.130·23-s − 0.352·25-s + 0.406·27-s − 0.121·29-s + 0.300·31-s − 0.252·33-s + 0.900·35-s − 0.426·37-s + 1.57·39-s + 1.04·41-s + 1.79·43-s − 1.02·45-s + 1.89·47-s + 0.252·49-s + 0.517·51-s + 1.32·53-s + 0.134·55-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

\(L(1)\) \(\approx\) \(2.570084994\)
\(L(\frac12)\) \(\approx\) \(2.570084994\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.60T + 3T^{2} \)
5 \( 1 + 1.79T + 5T^{2} \)
7 \( 1 + 2.96T + 7T^{2} \)
11 \( 1 + 0.556T + 11T^{2} \)
13 \( 1 - 3.76T + 13T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 - 1.17T + 19T^{2} \)
23 \( 1 - 0.626T + 23T^{2} \)
29 \( 1 + 0.654T + 29T^{2} \)
31 \( 1 - 1.67T + 31T^{2} \)
37 \( 1 + 2.59T + 37T^{2} \)
41 \( 1 - 6.68T + 41T^{2} \)
43 \( 1 - 11.7T + 43T^{2} \)
47 \( 1 - 13.0T + 47T^{2} \)
53 \( 1 - 9.66T + 53T^{2} \)
59 \( 1 - 5.84T + 59T^{2} \)
61 \( 1 + 6.37T + 61T^{2} \)
67 \( 1 - 9.80T + 67T^{2} \)
71 \( 1 + 14.7T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 + 2.17T + 79T^{2} \)
83 \( 1 - 2.48T + 83T^{2} \)
89 \( 1 + 1.95T + 89T^{2} \)
97 \( 1 - 15.4T + 97T^{2} \)
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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

−8.547328665842078095766905394101, −7.66297346005149679432566842881, −7.36539881626559706859233045612, −6.33570599766962592161204869461, −5.56813113960552767593055668497, −4.12499251618407278954882696565, −3.78028809602676066942283726138, −3.04363404027642227704174061907, −2.30068956582852032439139812334, −0.852861542593058539916333028147, 0.852861542593058539916333028147, 2.30068956582852032439139812334, 3.04363404027642227704174061907, 3.78028809602676066942283726138, 4.12499251618407278954882696565, 5.56813113960552767593055668497, 6.33570599766962592161204869461, 7.36539881626559706859233045612, 7.66297346005149679432566842881, 8.547328665842078095766905394101

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