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Properties

Label 2-1344-56.27-c3-0-66
Degree $2$
Conductor $1344$
Sign $-0.908 + 0.417i$
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 − 20.7·5-s + (−11.8 − 14.2i)7-s − 9·9-s + 39.2·11-s + 36.2·13-s − 62.1i·15-s − 92.2i·17-s + 87.5i·19-s + (42.7 − 35.4i)21-s + 100. i·23-s + 303.·25-s − 27i·27-s + 36.2i·29-s − 264.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.85·5-s + (−0.638 − 0.770i)7-s − 0.333·9-s + 1.07·11-s + 0.773·13-s − 1.06i·15-s − 1.31i·17-s + 1.05i·19-s + (0.444 − 0.368i)21-s + 0.915i·23-s + 2.42·25-s − 0.192i·27-s + 0.231i·29-s − 1.53·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.908 + 0.417i$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ -0.908 + 0.417i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.1201715356\)
\(L(\frac12)\) \(\approx\) \(0.1201715356\)
\(L(\frac{5}{2})\) 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 \( 1 + (11.8 + 14.2i)T \)
good5 \( 1 + 20.7T + 125T^{2} \)
11 \( 1 - 39.2T + 1.33e3T^{2} \)
13 \( 1 - 36.2T + 2.19e3T^{2} \)
17 \( 1 + 92.2iT - 4.91e3T^{2} \)
19 \( 1 - 87.5iT - 6.85e3T^{2} \)
23 \( 1 - 100. iT - 1.21e4T^{2} \)
29 \( 1 - 36.2iT - 2.43e4T^{2} \)
31 \( 1 + 264.T + 2.97e4T^{2} \)
37 \( 1 + 229. iT - 5.06e4T^{2} \)
41 \( 1 - 337. iT - 6.89e4T^{2} \)
43 \( 1 - 368.T + 7.95e4T^{2} \)
47 \( 1 - 314.T + 1.03e5T^{2} \)
53 \( 1 + 108. iT - 1.48e5T^{2} \)
59 \( 1 + 185. iT - 2.05e5T^{2} \)
61 \( 1 - 163.T + 2.26e5T^{2} \)
67 \( 1 - 458.T + 3.00e5T^{2} \)
71 \( 1 - 584. iT - 3.57e5T^{2} \)
73 \( 1 + 1.09e3iT - 3.89e5T^{2} \)
79 \( 1 + 438. iT - 4.93e5T^{2} \)
83 \( 1 + 722. iT - 5.71e5T^{2} \)
89 \( 1 - 452. iT - 7.04e5T^{2} \)
97 \( 1 - 597. iT - 9.12e5T^{2} \)
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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

−9.049243179651750103704777154041, −7.946722851529436081577990311775, −7.37780789197693969839927698270, −6.61231712992513206067215066246, −5.42773523051843125642017393628, −4.14433661969903111212718149390, −3.86550882701356599453087738396, −3.15110456029625964284315117116, −1.09050495364364501778973346222, −0.03778545833426911310018503889, 1.06678269653506846200819971286, 2.56580012236382938107351321429, 3.70313397128079797465929800180, 4.13490966206653127603511564514, 5.55539422137677821874393413033, 6.55928383674531555519617676852, 7.04465956124609720184362109837, 8.077668802797393793150943104374, 8.687388857718352830994455739355, 9.191958942438383610102475638455

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