Properties

Label 2-1344-28.27-c3-0-69
Degree $2$
Conductor $1344$
Sign $-0.262 + 0.964i$
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 − 7.52i·5-s + (4.86 − 17.8i)7-s + 9·9-s + 29.4i·11-s + 5.35i·13-s + 22.5i·15-s − 66.3i·17-s − 22.3·19-s + (−14.5 + 53.6i)21-s − 33.9i·23-s + 68.4·25-s − 27·27-s + 133.·29-s + 323.·31-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.672i·5-s + (0.262 − 0.964i)7-s + 0.333·9-s + 0.806i·11-s + 0.114i·13-s + 0.388i·15-s − 0.946i·17-s − 0.270·19-s + (−0.151 + 0.557i)21-s − 0.308i·23-s + 0.547·25-s − 0.192·27-s + 0.853·29-s + 1.87·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.262 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.262 + 0.964i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.262 + 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.591396526\)
\(L(\frac12)\) \(\approx\) \(1.591396526\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.86 + 17.8i)T \)
good5 \( 1 + 7.52iT - 125T^{2} \)
11 \( 1 - 29.4iT - 1.33e3T^{2} \)
13 \( 1 - 5.35iT - 2.19e3T^{2} \)
17 \( 1 + 66.3iT - 4.91e3T^{2} \)
19 \( 1 + 22.3T + 6.85e3T^{2} \)
23 \( 1 + 33.9iT - 1.21e4T^{2} \)
29 \( 1 - 133.T + 2.43e4T^{2} \)
31 \( 1 - 323.T + 2.97e4T^{2} \)
37 \( 1 + 120.T + 5.06e4T^{2} \)
41 \( 1 + 140. iT - 6.89e4T^{2} \)
43 \( 1 - 19.0iT - 7.95e4T^{2} \)
47 \( 1 - 376.T + 1.03e5T^{2} \)
53 \( 1 - 441.T + 1.48e5T^{2} \)
59 \( 1 + 241.T + 2.05e5T^{2} \)
61 \( 1 - 130. iT - 2.26e5T^{2} \)
67 \( 1 - 627. iT - 3.00e5T^{2} \)
71 \( 1 - 808. iT - 3.57e5T^{2} \)
73 \( 1 + 417. iT - 3.89e5T^{2} \)
79 \( 1 + 214. iT - 4.93e5T^{2} \)
83 \( 1 + 639.T + 5.71e5T^{2} \)
89 \( 1 + 686. iT - 7.04e5T^{2} \)
97 \( 1 + 103. iT - 9.12e5T^{2} \)
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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

−8.994268768767099512522215137910, −8.202217314648113297329648832278, −7.18829430428545317957311544234, −6.73405746132215632131817339740, −5.51997270770620590431693672026, −4.61915728718129934827567887348, −4.24843478090826957558926320652, −2.70206541314125665420854480135, −1.31232507447489349344413108847, −0.48907393495055522748944554826, 1.03843903598833781471259177340, 2.37953788963474483980019547046, 3.27735791870576655059356371561, 4.47808576942380297012107999799, 5.45579507525349016678020269127, 6.19698357589171994327598300211, 6.75876042330701679250133316974, 8.013635782833024272724896498023, 8.545492921257936318392388724135, 9.509467793902934685375852432489

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