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Properties

Label 2-1344-28.27-c3-0-83
Degree $2$
Conductor $1344$
Sign $-0.702 + 0.711i$
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 − 19.8i·5-s + (13.1 + 13.0i)7-s + 9·9-s + 11.9i·11-s + 82.5i·13-s − 59.4i·15-s − 100. i·17-s − 104.·19-s + (39.5 + 39.0i)21-s − 144. i·23-s − 268.·25-s + 27·27-s − 92.0·29-s + 195.·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.77i·5-s + (0.711 + 0.702i)7-s + 0.333·9-s + 0.327i·11-s + 1.76i·13-s − 1.02i·15-s − 1.43i·17-s − 1.25·19-s + (0.410 + 0.405i)21-s − 1.31i·23-s − 2.14·25-s + 0.192·27-s − 0.589·29-s + 1.13·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.702 + 0.711i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.702 + 0.711i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.871056360\)
\(L(\frac12)\) \(\approx\) \(1.871056360\)
\(L(\frac{5}{2})\) 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 \( 1 + (-13.1 - 13.0i)T \)
good5 \( 1 + 19.8iT - 125T^{2} \)
11 \( 1 - 11.9iT - 1.33e3T^{2} \)
13 \( 1 - 82.5iT - 2.19e3T^{2} \)
17 \( 1 + 100. iT - 4.91e3T^{2} \)
19 \( 1 + 104.T + 6.85e3T^{2} \)
23 \( 1 + 144. iT - 1.21e4T^{2} \)
29 \( 1 + 92.0T + 2.43e4T^{2} \)
31 \( 1 - 195.T + 2.97e4T^{2} \)
37 \( 1 + 281.T + 5.06e4T^{2} \)
41 \( 1 + 435. iT - 6.89e4T^{2} \)
43 \( 1 + 362. iT - 7.95e4T^{2} \)
47 \( 1 + 35.6T + 1.03e5T^{2} \)
53 \( 1 - 68.5T + 1.48e5T^{2} \)
59 \( 1 - 460.T + 2.05e5T^{2} \)
61 \( 1 - 443. iT - 2.26e5T^{2} \)
67 \( 1 + 867. iT - 3.00e5T^{2} \)
71 \( 1 + 577. iT - 3.57e5T^{2} \)
73 \( 1 + 590. iT - 3.89e5T^{2} \)
79 \( 1 + 282. iT - 4.93e5T^{2} \)
83 \( 1 + 816.T + 5.71e5T^{2} \)
89 \( 1 + 849. iT - 7.04e5T^{2} \)
97 \( 1 - 711. iT - 9.12e5T^{2} \)
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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

−8.825270246269434391520575902604, −8.562774518341902777543174970761, −7.46730410581866494564739974021, −6.51676124511038828071748087883, −5.27037320368769444334812380146, −4.65066494189740140981403002651, −4.08017490188011179698019969432, −2.28249438114065743443590517416, −1.74615456669847793464476332042, −0.36842723141161152802088096931, 1.40257968362707814730668996829, 2.59030335362734091882471214015, 3.39757213147670954737857503692, 4.09050218502853643290649332932, 5.52517432638991364886567244048, 6.39472723296248537437351541628, 7.17479451573563380660743077472, 8.089461389612285956056574621905, 8.266530146389980178270761991167, 9.939537488881821878774848412445

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