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Properties

Label 2-1344-28.27-c3-0-39
Degree $2$
Conductor $1344$
Sign $0.755 - 0.654i$
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 + 13.8i·5-s + (−14 + 12.1i)7-s + 9·9-s + 3.46i·11-s + 13.8i·13-s − 41.5i·15-s − 76.2i·17-s + 52·19-s + (42 − 36.3i)21-s − 114. i·23-s − 66.9·25-s − 27·27-s + 246·29-s − 116·31-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.23i·5-s + (−0.755 + 0.654i)7-s + 0.333·9-s + 0.0949i·11-s + 0.295i·13-s − 0.715i·15-s − 1.08i·17-s + 0.627·19-s + (0.436 − 0.377i)21-s − 1.03i·23-s − 0.535·25-s − 0.192·27-s + 1.57·29-s − 0.672·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.755 - 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.755 - 0.654i$
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: \((2,\ 1344,\ (\ :3/2),\ 0.755 - 0.654i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.456384685\)
\(L(\frac12)\) \(\approx\) \(1.456384685\)
\(L(\frac{5}{2})\) 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 \( 1 + (14 - 12.1i)T \)
good5 \( 1 - 13.8iT - 125T^{2} \)
11 \( 1 - 3.46iT - 1.33e3T^{2} \)
13 \( 1 - 13.8iT - 2.19e3T^{2} \)
17 \( 1 + 76.2iT - 4.91e3T^{2} \)
19 \( 1 - 52T + 6.85e3T^{2} \)
23 \( 1 + 114. iT - 1.21e4T^{2} \)
29 \( 1 - 246T + 2.43e4T^{2} \)
31 \( 1 + 116T + 2.97e4T^{2} \)
37 \( 1 - 314T + 5.06e4T^{2} \)
41 \( 1 + 270. iT - 6.89e4T^{2} \)
43 \( 1 + 377. iT - 7.95e4T^{2} \)
47 \( 1 - 192T + 1.03e5T^{2} \)
53 \( 1 - 150T + 1.48e5T^{2} \)
59 \( 1 + 204T + 2.05e5T^{2} \)
61 \( 1 - 581. iT - 2.26e5T^{2} \)
67 \( 1 - 509. iT - 3.00e5T^{2} \)
71 \( 1 - 814. iT - 3.57e5T^{2} \)
73 \( 1 + 124. iT - 3.89e5T^{2} \)
79 \( 1 + 1.37e3iT - 4.93e5T^{2} \)
83 \( 1 + 252T + 5.71e5T^{2} \)
89 \( 1 - 214. iT - 7.04e5T^{2} \)
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

−9.452234465762091908446567591500, −8.652406616150924058399240937963, −7.33745949963243945207960241956, −6.87511693274788987505882236444, −6.12857873746104536084608048506, −5.31721563981983760977141866164, −4.17891732119555044015973837002, −2.99656696434686811478364497946, −2.39975156006199665194600330098, −0.62065161231511503064279285862, 0.67333436113738245219914469059, 1.40660716818317327457176323257, 3.12320299108044453136204534024, 4.15703696396978817776868861122, 4.90679729083314797923899284837, 5.84295737597222791708814085882, 6.51028380038808174789634467548, 7.62565449783769938743984017855, 8.268981055397111286008859949177, 9.340679327789296855640000994584

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