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Properties

Label 2-1344-28.27-c3-0-93
Degree $2$
Conductor $1344$
Sign $-0.812 + 0.583i$
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 − 16.6i·5-s + (15.0 − 10.8i)7-s + 9·9-s − 64.0i·11-s + 28.6i·13-s − 49.9i·15-s − 82.9i·17-s + 17.1·19-s + (45.1 − 32.4i)21-s − 95.0i·23-s − 152.·25-s + 27·27-s + 197.·29-s − 153.·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.48i·5-s + (0.812 − 0.583i)7-s + 0.333·9-s − 1.75i·11-s + 0.612i·13-s − 0.859i·15-s − 1.18i·17-s + 0.206·19-s + (0.468 − 0.336i)21-s − 0.861i·23-s − 1.21·25-s + 0.192·27-s + 1.26·29-s − 0.889·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.812 + 0.583i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.812 + 0.583i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.857596198\)
\(L(\frac12)\) \(\approx\) \(2.857596198\)
\(L(\frac{5}{2})\) 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 \( 1 + (-15.0 + 10.8i)T \)
good5 \( 1 + 16.6iT - 125T^{2} \)
11 \( 1 + 64.0iT - 1.33e3T^{2} \)
13 \( 1 - 28.6iT - 2.19e3T^{2} \)
17 \( 1 + 82.9iT - 4.91e3T^{2} \)
19 \( 1 - 17.1T + 6.85e3T^{2} \)
23 \( 1 + 95.0iT - 1.21e4T^{2} \)
29 \( 1 - 197.T + 2.43e4T^{2} \)
31 \( 1 + 153.T + 2.97e4T^{2} \)
37 \( 1 + 10.7T + 5.06e4T^{2} \)
41 \( 1 + 41.1iT - 6.89e4T^{2} \)
43 \( 1 - 412. iT - 7.95e4T^{2} \)
47 \( 1 - 477.T + 1.03e5T^{2} \)
53 \( 1 - 35.2T + 1.48e5T^{2} \)
59 \( 1 + 494.T + 2.05e5T^{2} \)
61 \( 1 + 294. iT - 2.26e5T^{2} \)
67 \( 1 - 207. iT - 3.00e5T^{2} \)
71 \( 1 - 534. iT - 3.57e5T^{2} \)
73 \( 1 - 582. iT - 3.89e5T^{2} \)
79 \( 1 + 311. iT - 4.93e5T^{2} \)
83 \( 1 - 1.31e3T + 5.71e5T^{2} \)
89 \( 1 - 616. iT - 7.04e5T^{2} \)
97 \( 1 + 104. iT - 9.12e5T^{2} \)
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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

−8.844468156983489300249320696663, −8.274680870471862245090357471177, −7.58373692277883230955529509418, −6.44703895530488041407078057950, −5.31737074370486389852214764000, −4.66939131172424665149638833265, −3.84035616204271200291867588550, −2.62989829955243189398056741909, −1.23714614287712881403549019545, −0.63049550018854547827610509681, 1.69894008276625758258700398504, 2.37578725266548372016028464945, 3.36043456058905665685862433447, 4.34466774390550787299808106206, 5.41395951420195196000869039324, 6.41387949290572024669285124715, 7.36317802883778463684624529253, 7.70673289281464006825072659740, 8.736800534011775190464666294439, 9.623469593599193453584956809859

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