Properties

Degree $2$
Conductor $1344$
Sign $0.387 - 0.921i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 1.22i)3-s + 2.44·5-s + (1 + 2.44i)7-s + 2.99i·9-s − 2.44i·13-s + (2.99 + 2.99i)15-s + 4.89·17-s − 2.44i·19-s + (−1.77 + 4.22i)21-s + 6i·23-s + 0.999·25-s + (−3.67 + 3.67i)27-s − 6i·29-s + (2.44 + 5.99i)35-s + 2·37-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s + 1.09·5-s + (0.377 + 0.925i)7-s + 0.999i·9-s − 0.679i·13-s + (0.774 + 0.774i)15-s + 1.18·17-s − 0.561i·19-s + (−0.387 + 0.921i)21-s + 1.25i·23-s + 0.199·25-s + (−0.707 + 0.707i)27-s − 1.11i·29-s + (0.414 + 1.01i)35-s + 0.328·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.387 - 0.921i$
Motivic weight: \(1\)
Character: $\chi_{1344} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.387 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.741847515\)
\(L(\frac12)\) \(\approx\) \(2.741847515\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.22 - 1.22i)T \)
7 \( 1 + (-1 - 2.44i)T \)
good5 \( 1 - 2.44T + 5T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + 2.44iT - 13T^{2} \)
17 \( 1 - 4.89T + 17T^{2} \)
19 \( 1 + 2.44iT - 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + 4.89T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + 4.89T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 + 12.2iT - 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 9.79iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 2.44T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 4.89iT - 97T^{2} \)
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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

−9.489220541971416473657323253942, −9.324232321219682946150939364573, −8.146533330175561852106786406887, −7.68877592687873002278933325824, −6.19015941168699676005615599687, −5.48528843572093334465524675341, −4.87588054465440716632789979036, −3.50211278522476471249498454669, −2.63687544521747585790303227821, −1.69131032866680340528989610033, 1.16417571717573110167605445649, 1.99038919907473091136245303705, 3.16954350753048597522444517988, 4.20186088531844883930035889134, 5.37598881321753188306594533393, 6.36062411835278405477238570711, 6.99570394146391909433856776171, 7.86595516989425928118252501410, 8.573994528057141721048411346874, 9.512082642079016747906110700733

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