Properties

Label 2-1344-8.5-c3-0-66
Degree $2$
Conductor $1344$
Sign $-0.965 - 0.258i$
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  − 3i·3-s − 16.7i·5-s + 7·7-s − 9·9-s − 16.7i·11-s − 86.0i·13-s − 50.3·15-s + 87.5·17-s − 104. i·19-s − 21i·21-s − 171.·23-s − 157.·25-s + 27i·27-s − 88.8i·29-s + 53.1·31-s + ⋯
L(s)  = 1  − 0.577i·3-s − 1.50i·5-s + 0.377·7-s − 0.333·9-s − 0.459i·11-s − 1.83i·13-s − 0.867·15-s + 1.24·17-s − 1.26i·19-s − 0.218i·21-s − 1.55·23-s − 1.25·25-s + 0.192i·27-s − 0.568i·29-s + 0.307·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.965 - 0.258i$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ -0.965 - 0.258i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.976353844\)
\(L(\frac12)\) \(\approx\) \(1.976353844\)
\(L(\frac{5}{2})\) 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 \( 1 + 3iT \)
7 \( 1 - 7T \)
good5 \( 1 + 16.7iT - 125T^{2} \)
11 \( 1 + 16.7iT - 1.33e3T^{2} \)
13 \( 1 + 86.0iT - 2.19e3T^{2} \)
17 \( 1 - 87.5T + 4.91e3T^{2} \)
19 \( 1 + 104. iT - 6.85e3T^{2} \)
23 \( 1 + 171.T + 1.21e4T^{2} \)
29 \( 1 + 88.8iT - 2.43e4T^{2} \)
31 \( 1 - 53.1T + 2.97e4T^{2} \)
37 \( 1 - 215. iT - 5.06e4T^{2} \)
41 \( 1 - 396.T + 6.89e4T^{2} \)
43 \( 1 + 132. iT - 7.95e4T^{2} \)
47 \( 1 - 325.T + 1.03e5T^{2} \)
53 \( 1 + 674. iT - 1.48e5T^{2} \)
59 \( 1 - 254. iT - 2.05e5T^{2} \)
61 \( 1 + 815. iT - 2.26e5T^{2} \)
67 \( 1 - 616. iT - 3.00e5T^{2} \)
71 \( 1 - 41.3T + 3.57e5T^{2} \)
73 \( 1 + 7.03T + 3.89e5T^{2} \)
79 \( 1 - 418.T + 4.93e5T^{2} \)
83 \( 1 + 93.4iT - 5.71e5T^{2} \)
89 \( 1 + 88.6T + 7.04e5T^{2} \)
97 \( 1 - 379.T + 9.12e5T^{2} \)
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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

−8.498992962075921748906373038252, −8.127701412328784787071897773184, −7.48160243262038614285328438454, −6.05894833949761311484539635706, −5.47686008955671908676842844667, −4.74671034557739378104089223426, −3.54436152634923980173277832038, −2.35544105385072326868142228155, −0.989284910623583746390554197857, −0.53358382772039845266429721168, 1.66622182071861445474727056809, 2.62382270371013045931181497352, 3.78915500740565523165290574088, 4.29958502212448073093221341065, 5.70444071223968629661504281827, 6.31645633842506528337550377136, 7.33646434899791568135747827282, 7.84718997362870338403802272490, 9.079099114364069525145618269514, 9.803907417539335969485148393189

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