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Properties

Label 2-1344-8.5-c3-0-59
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 − 7.86i·5-s + 7·7-s − 9·9-s + 31.0i·11-s + 48.9i·13-s + 23.5·15-s − 43.0·17-s − 110. i·19-s + 21i·21-s − 170.·23-s + 63.1·25-s − 27i·27-s + 78.6i·29-s + 48.8·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.703i·5-s + 0.377·7-s − 0.333·9-s + 0.852i·11-s + 1.04i·13-s + 0.406·15-s − 0.614·17-s − 1.32i·19-s + 0.218i·21-s − 1.54·23-s + 0.505·25-s − 0.192i·27-s + 0.503i·29-s + 0.283·31-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

\(L(2)\) \(\approx\) \(0.3483406143\)
\(L(\frac12)\) \(\approx\) \(0.3483406143\)
\(L(\frac{5}{2})\) 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 \( 1 + 7.86iT - 125T^{2} \)
11 \( 1 - 31.0iT - 1.33e3T^{2} \)
13 \( 1 - 48.9iT - 2.19e3T^{2} \)
17 \( 1 + 43.0T + 4.91e3T^{2} \)
19 \( 1 + 110. iT - 6.85e3T^{2} \)
23 \( 1 + 170.T + 1.21e4T^{2} \)
29 \( 1 - 78.6iT - 2.43e4T^{2} \)
31 \( 1 - 48.8T + 2.97e4T^{2} \)
37 \( 1 + 6.95iT - 5.06e4T^{2} \)
41 \( 1 - 339.T + 6.89e4T^{2} \)
43 \( 1 + 222. iT - 7.95e4T^{2} \)
47 \( 1 + 314.T + 1.03e5T^{2} \)
53 \( 1 + 487. iT - 1.48e5T^{2} \)
59 \( 1 - 500. iT - 2.05e5T^{2} \)
61 \( 1 - 29.6iT - 2.26e5T^{2} \)
67 \( 1 + 380. iT - 3.00e5T^{2} \)
71 \( 1 - 12.5T + 3.57e5T^{2} \)
73 \( 1 + 857.T + 3.89e5T^{2} \)
79 \( 1 + 799.T + 4.93e5T^{2} \)
83 \( 1 + 178. iT - 5.71e5T^{2} \)
89 \( 1 + 1.02e3T + 7.04e5T^{2} \)
97 \( 1 + 360.T + 9.12e5T^{2} \)
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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

−9.004037105708234203274331921794, −8.362813544176503431441441824730, −7.28408767299145160251843499195, −6.50522164602326044788053939927, −5.34694958890525059222420538727, −4.54737715647784319580535536108, −4.11971994150390391463839631948, −2.58538007366982020400871090615, −1.58352276289248990775841277776, −0.079293174836687037594603445378, 1.22851299956587042258648470076, 2.43732342729136781587772185100, 3.30059927815394206696218613223, 4.36907537897628459588488056605, 5.77362995312518038533534790808, 6.08583305346251365868999110933, 7.16155478909306893749021330154, 8.061303310393719360103287421141, 8.347347384637160902534154275906, 9.635256630646100638629008557391

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