Properties

Degree $2$
Conductor $1344$
Sign $-0.968 + 0.250i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (1.5 − 2.59i)5-s + (−2.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−1.5 − 2.59i)11-s + 4·13-s − 3·15-s + (2 − 3.46i)19-s + (0.500 + 2.59i)21-s + (−2 − 3.46i)25-s + 0.999·27-s − 9·29-s + (−0.5 − 0.866i)31-s + (−1.5 + 2.59i)33-s + (−6 + 5.19i)35-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.670 − 1.16i)5-s + (−0.944 − 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.452 − 0.783i)11-s + 1.10·13-s − 0.774·15-s + (0.458 − 0.794i)19-s + (0.109 + 0.566i)21-s + (−0.400 − 0.692i)25-s + 0.192·27-s − 1.67·29-s + (−0.0898 − 0.155i)31-s + (−0.261 + 0.452i)33-s + (−1.01 + 0.878i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.968 + 0.250i$
Motivic weight: \(1\)
Character: $\chi_{1344} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.084396415\)
\(L(\frac12)\) \(\approx\) \(1.084396415\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good5 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.5 + 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5 - 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 9T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + T + 97T^{2} \)
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

−9.191027185931119777766546501830, −8.566238297858644265930923755764, −7.62411236990218427561809685546, −6.64494406104385811027769021964, −5.80939920583384302638292252103, −5.33894426610144068278579162544, −4.06400188275481559516034203982, −2.97864268332861178571832407954, −1.55445830857693967280399678653, −0.45760126575825555831842628499, 1.88787079251985058206090425438, 3.08691029457206792552022307801, 3.71443981500927125131399719682, 5.10711785909175866657311552068, 6.00759532064066943862210745351, 6.48730825145589396331878131390, 7.37428904978834398234973024076, 8.460211171309675713827260941914, 9.617811925300647502603449069179, 9.847129592906555769621365340945

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