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Properties

Label 2-1344-28.27-c3-0-82
Degree $2$
Conductor $1344$
Sign $-0.105 + 0.994i$
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 + 19.4i·5-s + (1.95 − 18.4i)7-s + 9·9-s − 24.6i·11-s + 5.25i·13-s + 58.3i·15-s − 80.8i·17-s + 86.5·19-s + (5.85 − 55.2i)21-s − 108. i·23-s − 253.·25-s + 27·27-s − 278.·29-s − 116.·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.74i·5-s + (0.105 − 0.994i)7-s + 0.333·9-s − 0.675i·11-s + 0.112i·13-s + 1.00i·15-s − 1.15i·17-s + 1.04·19-s + (0.0608 − 0.574i)21-s − 0.982i·23-s − 2.02·25-s + 0.192·27-s − 1.78·29-s − 0.675·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.105 + 0.994i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.105 + 0.994i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.450700098\)
\(L(\frac12)\) \(\approx\) \(1.450700098\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.95 + 18.4i)T \)
good5 \( 1 - 19.4iT - 125T^{2} \)
11 \( 1 + 24.6iT - 1.33e3T^{2} \)
13 \( 1 - 5.25iT - 2.19e3T^{2} \)
17 \( 1 + 80.8iT - 4.91e3T^{2} \)
19 \( 1 - 86.5T + 6.85e3T^{2} \)
23 \( 1 + 108. iT - 1.21e4T^{2} \)
29 \( 1 + 278.T + 2.43e4T^{2} \)
31 \( 1 + 116.T + 2.97e4T^{2} \)
37 \( 1 + 53.5T + 5.06e4T^{2} \)
41 \( 1 - 303. iT - 6.89e4T^{2} \)
43 \( 1 + 176. iT - 7.95e4T^{2} \)
47 \( 1 + 102.T + 1.03e5T^{2} \)
53 \( 1 + 185.T + 1.48e5T^{2} \)
59 \( 1 + 732.T + 2.05e5T^{2} \)
61 \( 1 + 443. iT - 2.26e5T^{2} \)
67 \( 1 - 166. iT - 3.00e5T^{2} \)
71 \( 1 + 378. iT - 3.57e5T^{2} \)
73 \( 1 + 664. iT - 3.89e5T^{2} \)
79 \( 1 + 737. iT - 4.93e5T^{2} \)
83 \( 1 + 913.T + 5.71e5T^{2} \)
89 \( 1 + 1.49e3iT - 7.04e5T^{2} \)
97 \( 1 - 1.44e3iT - 9.12e5T^{2} \)
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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

−9.205010451630972143433142553700, −7.916029441318213373707468488010, −7.37656775721532013813125422193, −6.82242288173278787844630166027, −5.89111989604634213558978093622, −4.60528349168585775412082428315, −3.39667963437992235738947457193, −3.11672011682146466946765953367, −1.85767747163481625034330247040, −0.28696972759280367039796915320, 1.37291519337440856662016723086, 1.98637130736667604418743661914, 3.46252432452023922442269754626, 4.37662939570822270721754805306, 5.36636786957306577974858919310, 5.76179990247141038280885316925, 7.30932571992337373020681761810, 8.021846575228413865498659701896, 8.731312897727351464541708538443, 9.354241456906493183598559730375

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