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Properties

Label 2-1344-28.27-c3-0-45
Degree $2$
Conductor $1344$
Sign $0.846 - 0.533i$
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 + 13.1i·5-s + (−9.87 − 15.6i)7-s + 9·9-s + 66.5i·11-s − 33.0i·13-s + 39.3i·15-s − 86.2i·17-s − 21.0·19-s + (−29.6 − 47.0i)21-s − 218. i·23-s − 46.7·25-s + 27·27-s + 133.·29-s + 324.·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.17i·5-s + (−0.533 − 0.846i)7-s + 0.333·9-s + 1.82i·11-s − 0.705i·13-s + 0.676i·15-s − 1.23i·17-s − 0.253·19-s + (−0.307 − 0.488i)21-s − 1.98i·23-s − 0.374·25-s + 0.192·27-s + 0.854·29-s + 1.87·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.846 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.846 - 0.533i$
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: \((2,\ 1344,\ (\ :3/2),\ 0.846 - 0.533i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.505032567\)
\(L(\frac12)\) \(\approx\) \(2.505032567\)
\(L(\frac{5}{2})\) 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 \( 1 + (9.87 + 15.6i)T \)
good5 \( 1 - 13.1iT - 125T^{2} \)
11 \( 1 - 66.5iT - 1.33e3T^{2} \)
13 \( 1 + 33.0iT - 2.19e3T^{2} \)
17 \( 1 + 86.2iT - 4.91e3T^{2} \)
19 \( 1 + 21.0T + 6.85e3T^{2} \)
23 \( 1 + 218. iT - 1.21e4T^{2} \)
29 \( 1 - 133.T + 2.43e4T^{2} \)
31 \( 1 - 324.T + 2.97e4T^{2} \)
37 \( 1 - 73.1T + 5.06e4T^{2} \)
41 \( 1 - 309. iT - 6.89e4T^{2} \)
43 \( 1 + 28.7iT - 7.95e4T^{2} \)
47 \( 1 - 162.T + 1.03e5T^{2} \)
53 \( 1 + 575.T + 1.48e5T^{2} \)
59 \( 1 - 861.T + 2.05e5T^{2} \)
61 \( 1 - 367. iT - 2.26e5T^{2} \)
67 \( 1 - 818. iT - 3.00e5T^{2} \)
71 \( 1 + 407. iT - 3.57e5T^{2} \)
73 \( 1 - 515. iT - 3.89e5T^{2} \)
79 \( 1 - 1.13e3iT - 4.93e5T^{2} \)
83 \( 1 - 941.T + 5.71e5T^{2} \)
89 \( 1 + 628. iT - 7.04e5T^{2} \)
97 \( 1 - 522. iT - 9.12e5T^{2} \)
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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

−9.609415105457077760489019192339, −8.370851825946561523371741168306, −7.56184039709303624706053786359, −6.81353525147899096880613307103, −6.51652215456981727495324421020, −4.79664468274271950081430628535, −4.19481462915674364717215927401, −2.84324157304882978686401084386, −2.54891342727287976148801061766, −0.822176065255670615718126840803, 0.73265700845245509930487111158, 1.82019139321754952255368761688, 3.08763620201596760912801244103, 3.85942354368716894660086750776, 4.99195516178100282687141592607, 5.88489825733795036963921700011, 6.48763471875105174570263941161, 7.969920067199842991868136698007, 8.468460025155711571470850420932, 9.021219311824034380322209853961

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