Properties

Degree $2$
Conductor $1344$
Sign $0.921 + 0.387i$
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 + (−4.22 − 1.77i)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 + (−0.921 − 0.387i)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

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.7504822354\)
\(L(\frac12)\) \(\approx\) \(0.7504822354\)
\(L(\frac{3}{2})\) 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

−9.481241516776044429590806128221, −8.761007830944461723965273949515, −8.123775954297019330792468059475, −7.01568126267459643881507341503, −6.19284776678851533764778476548, −5.18130993612111320894790478933, −4.55927359551235072689699061888, −3.63810058301232834793265164490, −2.47948362807726104950042591516, −0.45219101148943517851934378369, 0.929404761008709542750864254335, 2.23815481024385545411522403883, 3.94434655622340833293609367367, 4.36665652472949880544494838885, 5.55127204461544351173617560034, 6.53973916320152757045221116717, 7.35541669480826989514010200756, 7.74112112238123139826249501989, 8.620467373295022166664944385495, 9.768837083353615108442908293460

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