Properties

Label 2-1344-28.27-c3-0-73
Degree $2$
Conductor $1344$
Sign $-0.133 + 0.991i$
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 − 14.9i·5-s + (18.3 + 2.46i)7-s + 9·9-s − 21.4i·11-s − 66.7i·13-s + 44.8i·15-s + 66.3i·17-s + 126.·19-s + (−55.0 − 7.39i)21-s − 153. i·23-s − 98.6·25-s − 27·27-s + 198.·29-s + 273.·31-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.33i·5-s + (0.991 + 0.133i)7-s + 0.333·9-s − 0.588i·11-s − 1.42i·13-s + 0.772i·15-s + 0.946i·17-s + 1.53·19-s + (−0.572 − 0.0768i)21-s − 1.39i·23-s − 0.788·25-s − 0.192·27-s + 1.27·29-s + 1.58·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.133 + 0.991i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.133 + 0.991i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.245137752\)
\(L(\frac12)\) \(\approx\) \(2.245137752\)
\(L(\frac{5}{2})\) 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 \( 1 + (-18.3 - 2.46i)T \)
good5 \( 1 + 14.9iT - 125T^{2} \)
11 \( 1 + 21.4iT - 1.33e3T^{2} \)
13 \( 1 + 66.7iT - 2.19e3T^{2} \)
17 \( 1 - 66.3iT - 4.91e3T^{2} \)
19 \( 1 - 126.T + 6.85e3T^{2} \)
23 \( 1 + 153. iT - 1.21e4T^{2} \)
29 \( 1 - 198.T + 2.43e4T^{2} \)
31 \( 1 - 273.T + 2.97e4T^{2} \)
37 \( 1 + 135.T + 5.06e4T^{2} \)
41 \( 1 + 89.7iT - 6.89e4T^{2} \)
43 \( 1 - 115. iT - 7.95e4T^{2} \)
47 \( 1 + 74.7T + 1.03e5T^{2} \)
53 \( 1 - 484.T + 1.48e5T^{2} \)
59 \( 1 - 410.T + 2.05e5T^{2} \)
61 \( 1 - 753. iT - 2.26e5T^{2} \)
67 \( 1 + 684. iT - 3.00e5T^{2} \)
71 \( 1 - 266. iT - 3.57e5T^{2} \)
73 \( 1 + 210. iT - 3.89e5T^{2} \)
79 \( 1 - 821. iT - 4.93e5T^{2} \)
83 \( 1 + 863.T + 5.71e5T^{2} \)
89 \( 1 - 1.02e3iT - 7.04e5T^{2} \)
97 \( 1 - 528. iT - 9.12e5T^{2} \)
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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

−8.625816908515491379007650199502, −8.441081925716352750665795942471, −7.62292390705796318294702480709, −6.33278335425932825587548178500, −5.42665400506332617365621682752, −5.01316202157929500540579549015, −4.11407161166439710880315461352, −2.73591878821541988889402287799, −1.16091216616283108913081263569, −0.73810352254368675628155485806, 1.13124048951794037742921515908, 2.24140957019965703534617204332, 3.33943108281637257896437101449, 4.50692660888431360400559875751, 5.18866506421853502505306220274, 6.30741554766933637478461115088, 7.20403801663215109097230733215, 7.36857920072013928373294103687, 8.651570607918312994705631976280, 9.782224584628269231040904501791

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