↑

Properties

Label 2-2268-63.47-c1-0-4
Degree $2$
Conductor $2268$
Sign $-0.527 - 0.849i$
Analytic cond. $18.1100$
Root an. cond. $4.25559$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.5 + 0.866i)7-s + (1.5 − 0.866i)13-s + (3 + 1.73i)19-s − 5·25-s + (1.5 + 0.866i)31-s + (−0.5 + 0.866i)37-s + (−6.5 + 11.2i)43-s + (5.5 − 4.33i)49-s + (−13.5 + 7.79i)61-s + (−5.5 + 9.52i)67-s + (−12 + 6.92i)73-s + (6.5 + 11.2i)79-s + (−3 + 3.46i)91-s + (−4.5 − 2.59i)97-s + 19.0i·103-s + ⋯
L(s)  = 1  + (−0.944 + 0.327i)7-s + (0.416 − 0.240i)13-s + (0.688 + 0.397i)19-s − 25-s + (0.269 + 0.155i)31-s + (−0.0821 + 0.142i)37-s + (−0.991 + 1.71i)43-s + (0.785 − 0.618i)49-s + (−1.72 + 0.997i)61-s + (−0.671 + 1.16i)67-s + (−1.40 + 0.810i)73-s + (0.731 + 1.26i)79-s + (−0.314 + 0.363i)91-s + (−0.456 − 0.263i)97-s + 1.87i·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2268\)    =    \(2^{2} \cdot 3^{4} \cdot 7\)
Sign: $-0.527 - 0.849i$
Analytic conductor: \(18.1100\)
Root analytic conductor: \(4.25559\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2268} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2268,\ (\ :1/2),\ -0.527 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8202770307\)
\(L(\frac12)\) \(\approx\) \(0.8202770307\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + (2.5 - 0.866i)T \)
good5 \( 1 + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-1.5 + 0.866i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-1.5 - 0.866i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.5 - 11.2i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (13.5 - 7.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (12 - 6.92i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.5 - 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.5 + 2.59i)T + (48.5 + 84.0i)T^{2} \)
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

−9.388864325818333886960166776184, −8.516716471578055284606641674931, −7.78355752262482640852595879017, −6.90960688753177451043535748898, −6.09330162027382121480573330436, −5.53710870705810893910005476387, −4.39703787894456168822371796383, −3.44504170752310122390882633560, −2.70920893319427508277864287185, −1.33788737859811181903933735626, 0.28747563188217364963795253239, 1.76466625430330174180520404698, 3.04395739920776348433477450468, 3.73698883087402149560727063572, 4.70675584954683724076283224140, 5.73974845059926819904398224485, 6.42051192266612408718541967302, 7.20010452094821171349411770737, 7.906448417093133968886173049897, 8.930719900438780983262154241068

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