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Properties

Label 2-1344-28.27-c3-0-76
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

\(L(2)\) \(\approx\) \(1.808453704\)
\(L(\frac12)\) \(\approx\) \(1.808453704\)
\(L(\frac{5}{2})\) 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

−8.996355081521786207358894635897, −8.439115855518333061613837183062, −7.49465846248841026963027598881, −6.55058343570103769090172677985, −5.50760218843856002363540393489, −4.91894690402429978130214196702, −3.61406234362829064397381788106, −2.82817607839762233793645514779, −1.70886219594804867808869809872, −0.38421126551438276588426226576, 1.22877301237552752752414669866, 2.38981859193585531062676115500, 3.44821664774120722684450935559, 4.12057373921762624991625816604, 5.20547468943233289391147741665, 6.51879641561279555913152225395, 7.03121659354392157491147679617, 7.76930851973165972083844076930, 8.629791110185959452228377838460, 9.585963177524805868688974427165

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