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Properties

Label 2-1344-8.5-c3-0-18
Degree $2$
Conductor $1344$
Sign $0.707 - 0.707i$
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 − 15.6i·5-s − 7·7-s − 9·9-s + 13.3i·11-s − 7.74i·13-s + 46.8·15-s − 25.3·17-s + 71.9i·19-s − 21i·21-s − 23.6·23-s − 118.·25-s − 27i·27-s + 9.49i·29-s + 74.7·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.39i·5-s − 0.377·7-s − 0.333·9-s + 0.366i·11-s − 0.165i·13-s + 0.806·15-s − 0.361·17-s + 0.869i·19-s − 0.218i·21-s − 0.214·23-s − 0.951·25-s − 0.192i·27-s + 0.0607i·29-s + 0.433·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.707 - 0.707i$
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: \((2,\ 1344,\ (\ :3/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.474071164\)
\(L(\frac12)\) \(\approx\) \(1.474071164\)
\(L(\frac{5}{2})\) 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 \( 1 + 15.6iT - 125T^{2} \)
11 \( 1 - 13.3iT - 1.33e3T^{2} \)
13 \( 1 + 7.74iT - 2.19e3T^{2} \)
17 \( 1 + 25.3T + 4.91e3T^{2} \)
19 \( 1 - 71.9iT - 6.85e3T^{2} \)
23 \( 1 + 23.6T + 1.21e4T^{2} \)
29 \( 1 - 9.49iT - 2.43e4T^{2} \)
31 \( 1 - 74.7T + 2.97e4T^{2} \)
37 \( 1 + 34.2iT - 5.06e4T^{2} \)
41 \( 1 + 406.T + 6.89e4T^{2} \)
43 \( 1 - 141. iT - 7.95e4T^{2} \)
47 \( 1 - 127.T + 1.03e5T^{2} \)
53 \( 1 + 106. iT - 1.48e5T^{2} \)
59 \( 1 - 65.5iT - 2.05e5T^{2} \)
61 \( 1 - 35.5iT - 2.26e5T^{2} \)
67 \( 1 - 606. iT - 3.00e5T^{2} \)
71 \( 1 - 921.T + 3.57e5T^{2} \)
73 \( 1 - 705.T + 3.89e5T^{2} \)
79 \( 1 - 294.T + 4.93e5T^{2} \)
83 \( 1 - 291. iT - 5.71e5T^{2} \)
89 \( 1 - 0.646T + 7.04e5T^{2} \)
97 \( 1 - 1.35e3T + 9.12e5T^{2} \)
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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

−9.409639890832939785929976752805, −8.556346664090791289799453082580, −8.039142842284180046446778996233, −6.83670521622386997140566964121, −5.81492507918852381116809692736, −5.05218373085853155582088435685, −4.32248418486334951042440579109, −3.42978875653429663505599539866, −2.02833668579081431493455182242, −0.803153507669558084594089956797, 0.44459311519097580312742998148, 2.02710720029623164806076714069, 2.89142328240186905875591752782, 3.67364173845597576739704860297, 5.01327605970827895726832636550, 6.18919035607933934208868061075, 6.67466926091475395153307094054, 7.31246596045568126550389878138, 8.231841974740690446575550931189, 9.124363285558645129261765732087

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