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Properties

Label 2-1344-28.27-c3-0-30
Degree $2$
Conductor $1344$
Sign $0.808 - 0.587i$
Analytic cond. $79.2985$
Root an. cond. $8.90497$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s − 4.47i·5-s + (−14.9 + 10.8i)7-s + 9·9-s − 7.61i·11-s − 13.3i·13-s − 13.4i·15-s + 55.3i·17-s − 73.9·19-s + (−44.9 + 32.6i)21-s − 133. i·23-s + 104.·25-s + 27·27-s − 23.3·29-s + 241.·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.400i·5-s + (−0.808 + 0.587i)7-s + 0.333·9-s − 0.208i·11-s − 0.285i·13-s − 0.231i·15-s + 0.789i·17-s − 0.892·19-s + (−0.467 + 0.339i)21-s − 1.20i·23-s + 0.839·25-s + 0.192·27-s − 0.149·29-s + 1.39·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.587i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.808 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.808 - 0.587i$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (895, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ 0.808 - 0.587i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.107790692\)
\(L(\frac12)\) \(\approx\) \(2.107790692\)
\(L(\frac{5}{2})\) 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 \( 1 - 3T \)
7 \( 1 + (14.9 - 10.8i)T \)
good5 \( 1 + 4.47iT - 125T^{2} \)
11 \( 1 + 7.61iT - 1.33e3T^{2} \)
13 \( 1 + 13.3iT - 2.19e3T^{2} \)
17 \( 1 - 55.3iT - 4.91e3T^{2} \)
19 \( 1 + 73.9T + 6.85e3T^{2} \)
23 \( 1 + 133. iT - 1.21e4T^{2} \)
29 \( 1 + 23.3T + 2.43e4T^{2} \)
31 \( 1 - 241.T + 2.97e4T^{2} \)
37 \( 1 + 178.T + 5.06e4T^{2} \)
41 \( 1 - 494. iT - 6.89e4T^{2} \)
43 \( 1 - 72.6iT - 7.95e4T^{2} \)
47 \( 1 + 59.7T + 1.03e5T^{2} \)
53 \( 1 - 569.T + 1.48e5T^{2} \)
59 \( 1 - 59.5T + 2.05e5T^{2} \)
61 \( 1 - 629. iT - 2.26e5T^{2} \)
67 \( 1 + 599. iT - 3.00e5T^{2} \)
71 \( 1 - 407. iT - 3.57e5T^{2} \)
73 \( 1 + 680. iT - 3.89e5T^{2} \)
79 \( 1 - 1.08e3iT - 4.93e5T^{2} \)
83 \( 1 - 935.T + 5.71e5T^{2} \)
89 \( 1 + 12.1iT - 7.04e5T^{2} \)
97 \( 1 - 1.43e3iT - 9.12e5T^{2} \)
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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

−9.125005784348121522668416504967, −8.568750673883129576868806457107, −7.996301017925508097305242837334, −6.67198255379110814975927717100, −6.21957189967791443685670195714, −5.04778168226556337995496507955, −4.12544517513698805399836666511, −3.07800928671166172503745600104, −2.27445033549127407442467379334, −0.871249948841804482678563881374, 0.56019585927837726115214059239, 2.01641546427476331828217790003, 3.05264127812046707044826783414, 3.81289087216008729928569959537, 4.78736625878988657211461912917, 5.98394038083678762618743390438, 7.01734795374817364313883480036, 7.24763931888229223354004880530, 8.466849032237792785708797046323, 9.156518676712330286702239196393

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