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Properties

Label 2-1344-28.27-c3-0-81
Degree $2$
Conductor $1344$
Sign $0.113 + 0.993i$
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 + 17.7i·5-s + (2.09 + 18.4i)7-s + 9·9-s − 32.9i·11-s − 53.7i·13-s + 53.2i·15-s − 48.1i·17-s − 110.·19-s + (6.29 + 55.2i)21-s − 157. i·23-s − 189.·25-s + 27·27-s + 72.0·29-s + 184.·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.58i·5-s + (0.113 + 0.993i)7-s + 0.333·9-s − 0.902i·11-s − 1.14i·13-s + 0.915i·15-s − 0.686i·17-s − 1.33·19-s + (0.0653 + 0.573i)21-s − 1.43i·23-s − 1.51·25-s + 0.192·27-s + 0.461·29-s + 1.06·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.113 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.113 + 0.993i$
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: \((2,\ 1344,\ (\ :3/2),\ 0.113 + 0.993i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.167514210\)
\(L(\frac12)\) \(\approx\) \(1.167514210\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.09 - 18.4i)T \)
good5 \( 1 - 17.7iT - 125T^{2} \)
11 \( 1 + 32.9iT - 1.33e3T^{2} \)
13 \( 1 + 53.7iT - 2.19e3T^{2} \)
17 \( 1 + 48.1iT - 4.91e3T^{2} \)
19 \( 1 + 110.T + 6.85e3T^{2} \)
23 \( 1 + 157. iT - 1.21e4T^{2} \)
29 \( 1 - 72.0T + 2.43e4T^{2} \)
31 \( 1 - 184.T + 2.97e4T^{2} \)
37 \( 1 + 422.T + 5.06e4T^{2} \)
41 \( 1 + 346. iT - 6.89e4T^{2} \)
43 \( 1 + 198. iT - 7.95e4T^{2} \)
47 \( 1 + 29.0T + 1.03e5T^{2} \)
53 \( 1 + 682.T + 1.48e5T^{2} \)
59 \( 1 + 305.T + 2.05e5T^{2} \)
61 \( 1 - 172. iT - 2.26e5T^{2} \)
67 \( 1 + 109. iT - 3.00e5T^{2} \)
71 \( 1 - 741. iT - 3.57e5T^{2} \)
73 \( 1 + 752. iT - 3.89e5T^{2} \)
79 \( 1 + 449. iT - 4.93e5T^{2} \)
83 \( 1 + 262.T + 5.71e5T^{2} \)
89 \( 1 - 674. iT - 7.04e5T^{2} \)
97 \( 1 + 1.45e3iT - 9.12e5T^{2} \)
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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

−8.754216604492364240554747756005, −8.440757429300068469656701928840, −7.43680312082896876882963479466, −6.53012479430682082007667968438, −5.97522225204441312341622518156, −4.82114147057297906427267585411, −3.43990362302649742692934123467, −2.85006955708056536263964187199, −2.15875435522439263888222458067, −0.23106119466012009878045214356, 1.32617283325112991122281687570, 1.84400021772339842181958911996, 3.57096680307078031957121640451, 4.55697646875443388864928444933, 4.70831388448720910557161959773, 6.22986602071312588580918401972, 7.09005989544829992062765748120, 8.039379779653823774693193746504, 8.539256809524542985595234369948, 9.443052829037762139152230465470

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