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Properties

Label 2-1344-28.27-c3-0-65
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

\(L(2)\) \(\approx\) \(3.096938176\)
\(L(\frac12)\) \(\approx\) \(3.096938176\)
\(L(\frac{5}{2})\) 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

−8.825182923845168660962905998087, −8.395926179581432040461369469369, −8.009965443519900090799310916722, −6.63495826502382250587761592152, −5.73493218790902168977932490039, −4.71539319953946292687860823571, −4.27134229338648933868876605837, −2.80333992115501500798035612321, −1.82037552739781522090092323424, −0.800045534854660215341260278726, 0.995458627847810515670471980767, 2.31756275292643405135887190997, 3.08239546601472693535658463718, 4.11111524396321962121515738059, 4.95421450999669982829128978601, 6.29455299096144887572800920874, 6.99161717872348489286416765777, 7.67117527291064216357880532105, 8.372469770283260450172791875954, 9.435933731420810365812391764027

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