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Properties

Label 2-1344-28.27-c3-0-23
Degree $2$
Conductor $1344$
Sign $-0.874 - 0.485i$
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 + 12.4i·5-s + (8.98 − 16.1i)7-s + 9·9-s + 66.0i·11-s + 82.6i·13-s + 37.4i·15-s − 30.0i·17-s − 4.86·19-s + (26.9 − 48.5i)21-s + 113. i·23-s − 31.0·25-s + 27·27-s − 233.·29-s − 100.·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.11i·5-s + (0.485 − 0.874i)7-s + 0.333·9-s + 1.81i·11-s + 1.76i·13-s + 0.645i·15-s − 0.428i·17-s − 0.0587·19-s + (0.280 − 0.504i)21-s + 1.02i·23-s − 0.248·25-s + 0.192·27-s − 1.49·29-s − 0.583·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.874 - 0.485i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.874 - 0.485i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.874 - 0.485i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.930603029\)
\(L(\frac12)\) \(\approx\) \(1.930603029\)
\(L(\frac{5}{2})\) 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 \( 1 + (-8.98 + 16.1i)T \)
good5 \( 1 - 12.4iT - 125T^{2} \)
11 \( 1 - 66.0iT - 1.33e3T^{2} \)
13 \( 1 - 82.6iT - 2.19e3T^{2} \)
17 \( 1 + 30.0iT - 4.91e3T^{2} \)
19 \( 1 + 4.86T + 6.85e3T^{2} \)
23 \( 1 - 113. iT - 1.21e4T^{2} \)
29 \( 1 + 233.T + 2.43e4T^{2} \)
31 \( 1 + 100.T + 2.97e4T^{2} \)
37 \( 1 + 157.T + 5.06e4T^{2} \)
41 \( 1 + 433. iT - 6.89e4T^{2} \)
43 \( 1 + 217. iT - 7.95e4T^{2} \)
47 \( 1 - 328.T + 1.03e5T^{2} \)
53 \( 1 - 117.T + 1.48e5T^{2} \)
59 \( 1 + 384.T + 2.05e5T^{2} \)
61 \( 1 + 87.8iT - 2.26e5T^{2} \)
67 \( 1 + 305. iT - 3.00e5T^{2} \)
71 \( 1 - 1.07e3iT - 3.57e5T^{2} \)
73 \( 1 - 239. iT - 3.89e5T^{2} \)
79 \( 1 - 6.66iT - 4.93e5T^{2} \)
83 \( 1 - 443.T + 5.71e5T^{2} \)
89 \( 1 - 113. iT - 7.04e5T^{2} \)
97 \( 1 + 356. iT - 9.12e5T^{2} \)
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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

−9.543674417208536298913867225768, −8.982532347383198006098581098674, −7.55845791579690311248586800300, −7.18204838130793130878716474737, −6.80492820901275922884618718189, −5.27624233408310784747572644953, −4.20133666282030348838596199429, −3.69889205556272951477049564199, −2.23620792329668971019524695430, −1.69544769533290997065993365464, 0.38554468104604679311786846259, 1.40007522176774115279534254141, 2.72880606137278929402104326885, 3.52083873348983039716687651664, 4.76664582157064289937910528213, 5.57532215895605159136110483648, 6.09944608226347220209615960657, 7.71847083216026193140304644230, 8.287836689105202039832873041629, 8.720274356111121719598134354050

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