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Properties

Label 2-1344-28.27-c3-0-75
Degree $2$
Conductor $1344$
Sign $-0.820 + 0.571i$
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 − 4.15i·5-s + (−10.5 − 15.1i)7-s + 9·9-s − 26.7i·11-s − 36.8i·13-s + 12.4i·15-s − 41.0i·17-s + 110.·19-s + (31.7 + 45.5i)21-s − 69.5i·23-s + 107.·25-s − 27·27-s + 155.·29-s − 98.2·31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.371i·5-s + (−0.571 − 0.820i)7-s + 0.333·9-s − 0.732i·11-s − 0.786i·13-s + 0.214i·15-s − 0.585i·17-s + 1.33·19-s + (0.330 + 0.473i)21-s − 0.630i·23-s + 0.861·25-s − 0.192·27-s + 0.992·29-s − 0.569·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.820 + 0.571i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.820 + 0.571i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.400738166\)
\(L(\frac12)\) \(\approx\) \(1.400738166\)
\(L(\frac{5}{2})\) 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 \( 1 + (10.5 + 15.1i)T \)
good5 \( 1 + 4.15iT - 125T^{2} \)
11 \( 1 + 26.7iT - 1.33e3T^{2} \)
13 \( 1 + 36.8iT - 2.19e3T^{2} \)
17 \( 1 + 41.0iT - 4.91e3T^{2} \)
19 \( 1 - 110.T + 6.85e3T^{2} \)
23 \( 1 + 69.5iT - 1.21e4T^{2} \)
29 \( 1 - 155.T + 2.43e4T^{2} \)
31 \( 1 + 98.2T + 2.97e4T^{2} \)
37 \( 1 - 210.T + 5.06e4T^{2} \)
41 \( 1 - 0.600iT - 6.89e4T^{2} \)
43 \( 1 + 354. iT - 7.95e4T^{2} \)
47 \( 1 + 258.T + 1.03e5T^{2} \)
53 \( 1 - 274.T + 1.48e5T^{2} \)
59 \( 1 - 301.T + 2.05e5T^{2} \)
61 \( 1 + 469. iT - 2.26e5T^{2} \)
67 \( 1 - 605. iT - 3.00e5T^{2} \)
71 \( 1 + 497. iT - 3.57e5T^{2} \)
73 \( 1 - 429. iT - 3.89e5T^{2} \)
79 \( 1 + 1.04e3iT - 4.93e5T^{2} \)
83 \( 1 - 1.02e3T + 5.71e5T^{2} \)
89 \( 1 - 936. iT - 7.04e5T^{2} \)
97 \( 1 - 407. iT - 9.12e5T^{2} \)
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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

−8.979164689980329991069122622210, −8.038227524837861895788471252270, −7.20591981298288105105157699929, −6.45901905751095141392882884961, −5.50760295083640893255966765325, −4.81085597541859883148333324988, −3.68164176424155877618035216716, −2.82144836299089410064623869355, −1.01099424644434032176407854540, −0.45300537088677552823875417117, 1.22611591886913083740405032565, 2.43946476817830423671961537373, 3.45760175458971048124931648296, 4.60776225442937644557372120074, 5.45288633232486390938899604828, 6.33415346566287373952906771560, 6.94318650413687813436548727793, 7.83300406575069671709882862545, 8.947305877338339023947249202343, 9.630454542078337779130976873787

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