Properties

Label 2-4024-1.1-c1-0-71
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.70·3-s + 2.99·5-s − 2.81·7-s + 4.34·9-s − 3.12·11-s + 1.74·13-s − 8.11·15-s + 0.354·17-s − 0.768·19-s + 7.61·21-s + 2.62·23-s + 3.97·25-s − 3.63·27-s − 2.66·29-s + 7.34·31-s + 8.46·33-s − 8.41·35-s − 5.78·37-s − 4.71·39-s − 6.53·41-s + 11.2·43-s + 13.0·45-s − 11.5·47-s + 0.897·49-s − 0.960·51-s + 5.59·53-s − 9.35·55-s + ⋯
L(s)  = 1  − 1.56·3-s + 1.33·5-s − 1.06·7-s + 1.44·9-s − 0.941·11-s + 0.482·13-s − 2.09·15-s + 0.0859·17-s − 0.176·19-s + 1.66·21-s + 0.547·23-s + 0.794·25-s − 0.700·27-s − 0.494·29-s + 1.31·31-s + 1.47·33-s − 1.42·35-s − 0.950·37-s − 0.755·39-s − 1.02·41-s + 1.70·43-s + 1.93·45-s − 1.68·47-s + 0.128·49-s − 0.134·51-s + 0.768·53-s − 1.26·55-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: \(1\)
Selberg data: \((2,\ 4024,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.70T + 3T^{2} \)
5 \( 1 - 2.99T + 5T^{2} \)
7 \( 1 + 2.81T + 7T^{2} \)
11 \( 1 + 3.12T + 11T^{2} \)
13 \( 1 - 1.74T + 13T^{2} \)
17 \( 1 - 0.354T + 17T^{2} \)
19 \( 1 + 0.768T + 19T^{2} \)
23 \( 1 - 2.62T + 23T^{2} \)
29 \( 1 + 2.66T + 29T^{2} \)
31 \( 1 - 7.34T + 31T^{2} \)
37 \( 1 + 5.78T + 37T^{2} \)
41 \( 1 + 6.53T + 41T^{2} \)
43 \( 1 - 11.2T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 - 5.59T + 53T^{2} \)
59 \( 1 + 5.94T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 + 14.9T + 67T^{2} \)
71 \( 1 + 1.16T + 71T^{2} \)
73 \( 1 - 7.40T + 73T^{2} \)
79 \( 1 - 1.11T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 - 8.47T + 89T^{2} \)
97 \( 1 + 11.3T + 97T^{2} \)
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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

−8.016601672123811884528156228971, −6.85717812922946653219504769817, −6.51291702059053784375637274628, −5.79532319812295683534518234413, −5.38217680407997825376140465478, −4.61538561202636721773726728679, −3.36152086137341631286116464933, −2.37693239783064799291715364838, −1.20236544172396326431578701004, 0, 1.20236544172396326431578701004, 2.37693239783064799291715364838, 3.36152086137341631286116464933, 4.61538561202636721773726728679, 5.38217680407997825376140465478, 5.79532319812295683534518234413, 6.51291702059053784375637274628, 6.85717812922946653219504769817, 8.016601672123811884528156228971

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