Properties

Label 2-4024-1.1-c1-0-87
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.89·3-s + 3.16·5-s − 0.0867·7-s + 5.36·9-s + 1.65·11-s − 1.35·13-s − 9.14·15-s + 3.42·17-s + 0.324·19-s + 0.250·21-s − 8.51·23-s + 5.00·25-s − 6.84·27-s − 8.13·29-s − 6.31·31-s − 4.77·33-s − 0.274·35-s + 0.255·37-s + 3.90·39-s + 3.09·41-s − 4.19·43-s + 16.9·45-s − 0.838·47-s − 6.99·49-s − 9.91·51-s + 10.3·53-s + 5.22·55-s + ⋯
L(s)  = 1  − 1.66·3-s + 1.41·5-s − 0.0327·7-s + 1.78·9-s + 0.497·11-s − 0.374·13-s − 2.36·15-s + 0.831·17-s + 0.0745·19-s + 0.0547·21-s − 1.77·23-s + 1.00·25-s − 1.31·27-s − 1.51·29-s − 1.13·31-s − 0.830·33-s − 0.0463·35-s + 0.0420·37-s + 0.625·39-s + 0.483·41-s − 0.639·43-s + 2.53·45-s − 0.122·47-s − 0.998·49-s − 1.38·51-s + 1.42·53-s + 0.703·55-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 \( 1 + 2.89T + 3T^{2} \)
5 \( 1 - 3.16T + 5T^{2} \)
7 \( 1 + 0.0867T + 7T^{2} \)
11 \( 1 - 1.65T + 11T^{2} \)
13 \( 1 + 1.35T + 13T^{2} \)
17 \( 1 - 3.42T + 17T^{2} \)
19 \( 1 - 0.324T + 19T^{2} \)
23 \( 1 + 8.51T + 23T^{2} \)
29 \( 1 + 8.13T + 29T^{2} \)
31 \( 1 + 6.31T + 31T^{2} \)
37 \( 1 - 0.255T + 37T^{2} \)
41 \( 1 - 3.09T + 41T^{2} \)
43 \( 1 + 4.19T + 43T^{2} \)
47 \( 1 + 0.838T + 47T^{2} \)
53 \( 1 - 10.3T + 53T^{2} \)
59 \( 1 + 7.20T + 59T^{2} \)
61 \( 1 + 14.0T + 61T^{2} \)
67 \( 1 - 3.88T + 67T^{2} \)
71 \( 1 - 0.709T + 71T^{2} \)
73 \( 1 + 5.35T + 73T^{2} \)
79 \( 1 - 6.01T + 79T^{2} \)
83 \( 1 + 4.45T + 83T^{2} \)
89 \( 1 + 3.14T + 89T^{2} \)
97 \( 1 - 12.9T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.86058675992442422427499071180, −7.12009664810875844957777167327, −6.28192605630801080210083011956, −5.82266605729403717756176265572, −5.42069223194616852534245459065, −4.55084385373825514799223571168, −3.56518518095070166000309151018, −2.07634666591785978332282045381, −1.39652520189974452384154703911, 0, 1.39652520189974452384154703911, 2.07634666591785978332282045381, 3.56518518095070166000309151018, 4.55084385373825514799223571168, 5.42069223194616852534245459065, 5.82266605729403717756176265572, 6.28192605630801080210083011956, 7.12009664810875844957777167327, 7.86058675992442422427499071180

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