Properties

Degree $2$
Conductor $1344$
Sign $-0.386 - 0.922i$
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 + (0.5 − 0.866i)5-s + (−0.5 + 2.59i)7-s + (−0.499 + 0.866i)9-s + (−2.5 − 4.33i)11-s + 0.999·15-s + (2 + 3.46i)17-s + (−4 + 6.92i)19-s + (−2.5 + 0.866i)21-s + (−2 + 3.46i)23-s + (2 + 3.46i)25-s − 0.999·27-s + 5·29-s + (1.5 + 2.59i)31-s + (2.5 − 4.33i)33-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (−0.188 + 0.981i)7-s + (−0.166 + 0.288i)9-s + (−0.753 − 1.30i)11-s + 0.258·15-s + (0.485 + 0.840i)17-s + (−0.917 + 1.58i)19-s + (−0.545 + 0.188i)21-s + (−0.417 + 0.722i)23-s + (0.400 + 0.692i)25-s − 0.192·27-s + 0.928·29-s + (0.269 + 0.466i)31-s + (0.435 − 0.753i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.386 - 0.922i$
Motivic weight: \(1\)
Character: $\chi_{1344} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.354607567\)
\(L(\frac12)\) \(\approx\) \(1.354607567\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 2.59i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (4 - 6.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (3 - 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.5 - 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3 - 5.19i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.5 + 2.59i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7T + 83T^{2} \)
89 \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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

−9.964144387205616760757576587111, −8.829965770867876428337453515507, −8.476345920140551803579788351928, −7.77071468299319691845252194904, −6.18426617767866734472035526609, −5.77843658959272649656695787938, −4.91613989710912956000428964026, −3.66397785011834080852943130608, −2.91412062726369648038873762917, −1.61463216684733902763759458720, 0.52085885221818334410213551925, 2.18329792223318467754764112735, 2.90866236358056530313661970435, 4.30562658207975681101140838961, 4.96146644002096163886826022258, 6.40802998149937015491503655251, 6.94406259940964775861221891901, 7.57881509584073805797410453542, 8.437734929599080839022445815197, 9.472059038176369575204979014707

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