Properties

Degree $2$
Conductor $1344$
Sign $-0.605 + 0.795i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)3-s + (1 − 1.73i)5-s + (2.5 − 0.866i)7-s + (−0.499 + 0.866i)9-s + (−3 − 5.19i)11-s + 3·13-s − 1.99·15-s + (−2 − 3.46i)17-s + (−2.5 + 4.33i)19-s + (−2 − 1.73i)21-s + (2 − 3.46i)23-s + (0.500 + 0.866i)25-s + 0.999·27-s + 4·29-s + (−3.5 − 6.06i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.447 − 0.774i)5-s + (0.944 − 0.327i)7-s + (−0.166 + 0.288i)9-s + (−0.904 − 1.56i)11-s + 0.832·13-s − 0.516·15-s + (−0.485 − 0.840i)17-s + (−0.573 + 0.993i)19-s + (−0.436 − 0.377i)21-s + (0.417 − 0.722i)23-s + (0.100 + 0.173i)25-s + 0.192·27-s + 0.742·29-s + (−0.628 − 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.605 + 0.795i$
Motivic weight: \(1\)
Character: $\chi_{1344} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.530569608\)
\(L(\frac12)\) \(\approx\) \(1.530569608\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (3.5 + 6.06i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4.5 - 7.79i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4 - 6.92i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (7.5 + 12.9i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-4 + 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.075254067065326305929321751854, −8.366165907148431719707747166390, −7.995824062294730994288961410975, −6.79549591868374692899320467660, −5.85875984115795608758901075346, −5.27045819999909117227030853843, −4.36233391950209015420596156442, −3.01464359590303729352082080501, −1.70312533232172950039168046172, −0.65772956934483450366686268977, 1.78086551505235290708553481262, 2.66294329496635965840391789019, 4.02551398396858222818184726107, 4.90310028084826248753660019944, 5.59408299019937659414643583057, 6.68783967896404805240251239016, 7.30084185744788646178363752566, 8.446274712940039282657322985582, 9.046195217543252511145420893041, 10.18644630176911971846820965572

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