Properties

Label 2-1344-28.27-c3-0-15
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 + 6.58i·5-s + (15.1 + 10.5i)7-s + 9·9-s − 54.2i·11-s + 40.9i·13-s + 19.7i·15-s + 69.4i·17-s − 160.·19-s + (45.5 + 31.7i)21-s + 87.8i·23-s + 81.6·25-s + 27·27-s − 236.·29-s − 131.·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.589i·5-s + (0.820 + 0.571i)7-s + 0.333·9-s − 1.48i·11-s + 0.874i·13-s + 0.340i·15-s + 0.991i·17-s − 1.93·19-s + (0.473 + 0.330i)21-s + 0.796i·23-s + 0.652·25-s + 0.192·27-s − 1.51·29-s − 0.762·31-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

\(L(2)\) \(\approx\) \(1.538571206\)
\(L(\frac12)\) \(\approx\) \(1.538571206\)
\(L(\frac{5}{2})\) 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 \( 1 + (-15.1 - 10.5i)T \)
good5 \( 1 - 6.58iT - 125T^{2} \)
11 \( 1 + 54.2iT - 1.33e3T^{2} \)
13 \( 1 - 40.9iT - 2.19e3T^{2} \)
17 \( 1 - 69.4iT - 4.91e3T^{2} \)
19 \( 1 + 160.T + 6.85e3T^{2} \)
23 \( 1 - 87.8iT - 1.21e4T^{2} \)
29 \( 1 + 236.T + 2.43e4T^{2} \)
31 \( 1 + 131.T + 2.97e4T^{2} \)
37 \( 1 - 23.6T + 5.06e4T^{2} \)
41 \( 1 + 112. iT - 6.89e4T^{2} \)
43 \( 1 - 194. iT - 7.95e4T^{2} \)
47 \( 1 + 269.T + 1.03e5T^{2} \)
53 \( 1 - 120.T + 1.48e5T^{2} \)
59 \( 1 - 338.T + 2.05e5T^{2} \)
61 \( 1 - 267. iT - 2.26e5T^{2} \)
67 \( 1 + 275. iT - 3.00e5T^{2} \)
71 \( 1 - 270. iT - 3.57e5T^{2} \)
73 \( 1 - 1.23e3iT - 3.89e5T^{2} \)
79 \( 1 + 691. iT - 4.93e5T^{2} \)
83 \( 1 + 430.T + 5.71e5T^{2} \)
89 \( 1 + 1.22e3iT - 7.04e5T^{2} \)
97 \( 1 - 381. iT - 9.12e5T^{2} \)
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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

−9.322369585733411326502948238309, −8.634253391333340493127786041812, −8.210179426848016192800075748220, −7.16613390534826263350756400145, −6.25403625770143910961497062551, −5.53653831450275901785306553117, −4.25807276951648140331931508329, −3.50662416650347600072529323420, −2.36838051965825867150720877555, −1.55173698531274689653117922012, 0.29323056841764706620819588468, 1.63773188943924853646488162614, 2.44627995530951112986735261877, 3.88557211811057414382940491189, 4.62654548482810433982764526744, 5.25940229512426810567830852787, 6.69545856529276242841922356252, 7.40795985244579541968579647458, 8.126714723240908710016689816240, 8.844868033487025970790802776971

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