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Properties

Label 2-1344-8.5-c3-0-29
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 − 6.85i·5-s − 7·7-s − 9·9-s − 30.4i·11-s + 45.1i·13-s + 20.5·15-s − 99.4·17-s + 33.9i·19-s − 21i·21-s − 108.·23-s + 78.0·25-s − 27i·27-s − 187. i·29-s + 174.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.613i·5-s − 0.377·7-s − 0.333·9-s − 0.834i·11-s + 0.962i·13-s + 0.354·15-s − 1.41·17-s + 0.410i·19-s − 0.218i·21-s − 0.982·23-s + 0.624·25-s − 0.192i·27-s − 1.20i·29-s + 1.01·31-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

\(L(2)\) \(\approx\) \(1.567474282\)
\(L(\frac12)\) \(\approx\) \(1.567474282\)
\(L(\frac{5}{2})\) 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 \( 1 + 6.85iT - 125T^{2} \)
11 \( 1 + 30.4iT - 1.33e3T^{2} \)
13 \( 1 - 45.1iT - 2.19e3T^{2} \)
17 \( 1 + 99.4T + 4.91e3T^{2} \)
19 \( 1 - 33.9iT - 6.85e3T^{2} \)
23 \( 1 + 108.T + 1.21e4T^{2} \)
29 \( 1 + 187. iT - 2.43e4T^{2} \)
31 \( 1 - 174.T + 2.97e4T^{2} \)
37 \( 1 - 241. iT - 5.06e4T^{2} \)
41 \( 1 - 474.T + 6.89e4T^{2} \)
43 \( 1 - 480. iT - 7.95e4T^{2} \)
47 \( 1 + 516.T + 1.03e5T^{2} \)
53 \( 1 - 179. iT - 1.48e5T^{2} \)
59 \( 1 + 22.3iT - 2.05e5T^{2} \)
61 \( 1 + 932. iT - 2.26e5T^{2} \)
67 \( 1 + 665. iT - 3.00e5T^{2} \)
71 \( 1 - 129.T + 3.57e5T^{2} \)
73 \( 1 - 1.08e3T + 3.89e5T^{2} \)
79 \( 1 - 739.T + 4.93e5T^{2} \)
83 \( 1 + 81.6iT - 5.71e5T^{2} \)
89 \( 1 + 1.01e3T + 7.04e5T^{2} \)
97 \( 1 - 610.T + 9.12e5T^{2} \)
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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

−9.417893704682518155807175803158, −8.492671076803040225924345020026, −7.960924944738232885074949107901, −6.44567153035802077338381511312, −6.17574534731199876546547829995, −4.78940048027700171445232985507, −4.32117059117464479244191072477, −3.23114106868522322349704789828, −2.06929396745361167998015034377, −0.62080995340622742874727353249, 0.59648405649967787932009588861, 2.09607620541439449707375935722, 2.82381849637049346243794220960, 3.97909562669930376249915779541, 5.06085361676734679263870026698, 6.08770368044140628426625393253, 6.86531555174557124916644681721, 7.36814993512144837002251053399, 8.374023699187785960520731488637, 9.139088493735562043900984013023

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