Properties

Label 2-1344-56.27-c3-0-5
Degree $2$
Conductor $1344$
Sign $-0.417 - 0.908i$
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  − 3i·3-s − 20.7·5-s + (11.8 + 14.2i)7-s − 9·9-s − 39.2·11-s + 36.2·13-s + 62.1i·15-s − 92.2i·17-s − 87.5i·19-s + (42.7 − 35.4i)21-s − 100. i·23-s + 303.·25-s + 27i·27-s + 36.2i·29-s + 264.·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.85·5-s + (0.638 + 0.770i)7-s − 0.333·9-s − 1.07·11-s + 0.773·13-s + 1.06i·15-s − 1.31i·17-s − 1.05i·19-s + (0.444 − 0.368i)21-s − 0.915i·23-s + 2.42·25-s + 0.192i·27-s + 0.231i·29-s + 1.53·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.417 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.417 - 0.908i$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ -0.417 - 0.908i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1670904010\)
\(L(\frac12)\) \(\approx\) \(0.1670904010\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
7 \( 1 + (-11.8 - 14.2i)T \)
good5 \( 1 + 20.7T + 125T^{2} \)
11 \( 1 + 39.2T + 1.33e3T^{2} \)
13 \( 1 - 36.2T + 2.19e3T^{2} \)
17 \( 1 + 92.2iT - 4.91e3T^{2} \)
19 \( 1 + 87.5iT - 6.85e3T^{2} \)
23 \( 1 + 100. iT - 1.21e4T^{2} \)
29 \( 1 - 36.2iT - 2.43e4T^{2} \)
31 \( 1 - 264.T + 2.97e4T^{2} \)
37 \( 1 + 229. iT - 5.06e4T^{2} \)
41 \( 1 - 337. iT - 6.89e4T^{2} \)
43 \( 1 + 368.T + 7.95e4T^{2} \)
47 \( 1 + 314.T + 1.03e5T^{2} \)
53 \( 1 + 108. iT - 1.48e5T^{2} \)
59 \( 1 - 185. iT - 2.05e5T^{2} \)
61 \( 1 - 163.T + 2.26e5T^{2} \)
67 \( 1 + 458.T + 3.00e5T^{2} \)
71 \( 1 + 584. iT - 3.57e5T^{2} \)
73 \( 1 + 1.09e3iT - 3.89e5T^{2} \)
79 \( 1 - 438. iT - 4.93e5T^{2} \)
83 \( 1 - 722. iT - 5.71e5T^{2} \)
89 \( 1 - 452. iT - 7.04e5T^{2} \)
97 \( 1 - 597. iT - 9.12e5T^{2} \)
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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

−9.190886481703701696244682546849, −8.326585491410504035680693978721, −8.048415338099462068410598022265, −7.21969244160346972865928401435, −6.40630663594204314936552518210, −5.03142724505610399072843621913, −4.61870933591328217304161020801, −3.21913864501775287464267843720, −2.53262510177459466259288501169, −0.874119594320551316532076012751, 0.05252625903644881497038222007, 1.38332870538310031943018758759, 3.18967278825928807864052668006, 3.86734037062241388842175981337, 4.46876305075423389911106937869, 5.42330804928549163690187396044, 6.63871404298660382934732537964, 7.71276536269013796941703676432, 8.142818161152921780513585135899, 8.573140551813244472568165138050

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