Properties

Label 2-7448-1.1-c1-0-50
Degree $2$
Conductor $7448$
Sign $1$
Analytic cond. $59.4725$
Root an. cond. $7.71184$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.28·3-s + 1.50·5-s + 2.22·9-s + 2.10·11-s + 3.79·13-s − 3.43·15-s − 2.43·17-s − 19-s + 3.12·23-s − 2.74·25-s + 1.77·27-s − 8.00·29-s + 8.70·31-s − 4.80·33-s + 3.59·37-s − 8.68·39-s + 0.873·41-s + 5.89·43-s + 3.34·45-s + 4.76·47-s + 5.56·51-s − 11.4·53-s + 3.15·55-s + 2.28·57-s + 1.76·59-s + 12.6·61-s + 5.70·65-s + ⋯
L(s)  = 1  − 1.31·3-s + 0.671·5-s + 0.741·9-s + 0.634·11-s + 1.05·13-s − 0.886·15-s − 0.590·17-s − 0.229·19-s + 0.651·23-s − 0.548·25-s + 0.341·27-s − 1.48·29-s + 1.56·31-s − 0.836·33-s + 0.591·37-s − 1.39·39-s + 0.136·41-s + 0.898·43-s + 0.498·45-s + 0.695·47-s + 0.779·51-s − 1.56·53-s + 0.426·55-s + 0.302·57-s + 0.229·59-s + 1.62·61-s + 0.707·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7448\)    =    \(2^{3} \cdot 7^{2} \cdot 19\)
Sign: $1$
Analytic conductor: \(59.4725\)
Root analytic conductor: \(7.71184\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7448} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7448,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.463452617\)
\(L(\frac12)\) \(\approx\) \(1.463452617\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 + T \)
good3 \( 1 + 2.28T + 3T^{2} \)
5 \( 1 - 1.50T + 5T^{2} \)
11 \( 1 - 2.10T + 11T^{2} \)
13 \( 1 - 3.79T + 13T^{2} \)
17 \( 1 + 2.43T + 17T^{2} \)
23 \( 1 - 3.12T + 23T^{2} \)
29 \( 1 + 8.00T + 29T^{2} \)
31 \( 1 - 8.70T + 31T^{2} \)
37 \( 1 - 3.59T + 37T^{2} \)
41 \( 1 - 0.873T + 41T^{2} \)
43 \( 1 - 5.89T + 43T^{2} \)
47 \( 1 - 4.76T + 47T^{2} \)
53 \( 1 + 11.4T + 53T^{2} \)
59 \( 1 - 1.76T + 59T^{2} \)
61 \( 1 - 12.6T + 61T^{2} \)
67 \( 1 - 1.44T + 67T^{2} \)
71 \( 1 + 2.47T + 71T^{2} \)
73 \( 1 + 1.04T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 + 6.67T + 83T^{2} \)
89 \( 1 - 11.5T + 89T^{2} \)
97 \( 1 + 2.84T + 97T^{2} \)
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

−7.83473244283459634431063036256, −6.85946449305934148901324929319, −6.35084021967390760788047408751, −5.89886494791926528474643818372, −5.27659326122238505351731177661, −4.43249982950517200909708268389, −3.75691997088265340831970380138, −2.57894434228785023446464824389, −1.55389604807115672466591522309, −0.68998048756600089271000188327, 0.68998048756600089271000188327, 1.55389604807115672466591522309, 2.57894434228785023446464824389, 3.75691997088265340831970380138, 4.43249982950517200909708268389, 5.27659326122238505351731177661, 5.89886494791926528474643818372, 6.35084021967390760788047408751, 6.85946449305934148901324929319, 7.83473244283459634431063036256

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