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

\(L(1)\) \(\approx\) \(1.878115463\)
\(L(\frac12)\) \(\approx\) \(1.878115463\)
\(L(\frac{3}{2})\) 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

−9.661828778204744979263752974985, −9.061147635327130878898546492572, −8.476886523323609540548834333354, −7.05527745628719582604363649878, −6.60166636060455075415403691746, −5.33631258717304183008560037926, −4.71496711532365541581977928842, −3.69622942818843229777735936602, −2.62488581165896639895207039758, −1.31926011431250485134687295627, 0.846566050412001695512001384867, 2.29084217163218853439282551178, 3.43769372299176148214603165299, 3.87557989379165919252853137159, 5.78321394563122959727903046208, 6.35577448308589442350525109577, 6.69569905015701800945830599205, 7.996967516213156089272758234472, 8.618819852881984684475716843559, 9.454157858589829136293294036860

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