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Properties

Label 2-1344-28.27-c3-0-47
Degree $2$
Conductor $1344$
Sign $0.990 + 0.137i$
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 + 12.7i·5-s + (2.54 − 18.3i)7-s + 9·9-s + 2.42i·11-s + 26.6i·13-s − 38.1i·15-s + 10.6i·17-s + 30.0·19-s + (−7.62 + 55.0i)21-s − 120. i·23-s − 36.7·25-s − 27·27-s − 43.1·29-s − 6.30·31-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.13i·5-s + (0.137 − 0.990i)7-s + 0.333·9-s + 0.0664i·11-s + 0.568i·13-s − 0.656i·15-s + 0.151i·17-s + 0.363·19-s + (−0.0792 + 0.571i)21-s − 1.08i·23-s − 0.293·25-s − 0.192·27-s − 0.276·29-s − 0.0365·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.990 + 0.137i$
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: \((2,\ 1344,\ (\ :3/2),\ 0.990 + 0.137i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.544476718\)
\(L(\frac12)\) \(\approx\) \(1.544476718\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.54 + 18.3i)T \)
good5 \( 1 - 12.7iT - 125T^{2} \)
11 \( 1 - 2.42iT - 1.33e3T^{2} \)
13 \( 1 - 26.6iT - 2.19e3T^{2} \)
17 \( 1 - 10.6iT - 4.91e3T^{2} \)
19 \( 1 - 30.0T + 6.85e3T^{2} \)
23 \( 1 + 120. iT - 1.21e4T^{2} \)
29 \( 1 + 43.1T + 2.43e4T^{2} \)
31 \( 1 + 6.30T + 2.97e4T^{2} \)
37 \( 1 - 61.6T + 5.06e4T^{2} \)
41 \( 1 - 75.9iT - 6.89e4T^{2} \)
43 \( 1 + 212. iT - 7.95e4T^{2} \)
47 \( 1 + 330.T + 1.03e5T^{2} \)
53 \( 1 - 4.87T + 1.48e5T^{2} \)
59 \( 1 - 481.T + 2.05e5T^{2} \)
61 \( 1 - 236. iT - 2.26e5T^{2} \)
67 \( 1 + 661. iT - 3.00e5T^{2} \)
71 \( 1 - 547. iT - 3.57e5T^{2} \)
73 \( 1 + 968. iT - 3.89e5T^{2} \)
79 \( 1 + 64.6iT - 4.93e5T^{2} \)
83 \( 1 - 665.T + 5.71e5T^{2} \)
89 \( 1 + 384. iT - 7.04e5T^{2} \)
97 \( 1 + 1.22e3iT - 9.12e5T^{2} \)
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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

−9.440336723326206525994564056985, −8.267538197881054373526173112273, −7.33528458888468376433462462426, −6.79812998133483808273940935709, −6.14387199809863275547873602198, −4.94215988317733314564611013225, −4.08516638754438845835310817091, −3.14872482014210949979210301307, −1.90373801080887855666168541488, −0.56215320418525353302426359588, 0.74791824815257433906778081545, 1.77562109732306258731885950859, 3.10422809492645614444187939251, 4.34920776211480242338891462109, 5.35202174537006397091906826194, 5.54461061154136154613566310787, 6.69152607715745352463339325905, 7.81139637909353363748677280460, 8.456623291499505326734532138117, 9.314883899460762742812843276157

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