Properties

Label 2-1344-56.27-c3-0-56
Degree $2$
Conductor $1344$
Sign $0.913 - 0.406i$
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 + 12.5·5-s + (−18.2 + 2.88i)7-s − 9·9-s − 55.4·11-s + 92.1·13-s + 37.5i·15-s − 118. i·17-s + 155. i·19-s + (−8.66 − 54.8i)21-s − 125. i·23-s + 31.6·25-s − 27i·27-s − 131. i·29-s + 66.0·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.11·5-s + (−0.987 + 0.155i)7-s − 0.333·9-s − 1.51·11-s + 1.96·13-s + 0.646i·15-s − 1.68i·17-s + 1.87i·19-s + (−0.0900 − 0.570i)21-s − 1.13i·23-s + 0.253·25-s − 0.192i·27-s − 0.842i·29-s + 0.382·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.913 - 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.913 - 0.406i$
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: \((2,\ 1344,\ (\ :3/2),\ 0.913 - 0.406i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.216258767\)
\(L(\frac12)\) \(\approx\) \(2.216258767\)
\(L(\frac{5}{2})\) 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 \( 1 + (18.2 - 2.88i)T \)
good5 \( 1 - 12.5T + 125T^{2} \)
11 \( 1 + 55.4T + 1.33e3T^{2} \)
13 \( 1 - 92.1T + 2.19e3T^{2} \)
17 \( 1 + 118. iT - 4.91e3T^{2} \)
19 \( 1 - 155. iT - 6.85e3T^{2} \)
23 \( 1 + 125. iT - 1.21e4T^{2} \)
29 \( 1 + 131. iT - 2.43e4T^{2} \)
31 \( 1 - 66.0T + 2.97e4T^{2} \)
37 \( 1 - 147. iT - 5.06e4T^{2} \)
41 \( 1 + 20.3iT - 6.89e4T^{2} \)
43 \( 1 - 355.T + 7.95e4T^{2} \)
47 \( 1 - 79.5T + 1.03e5T^{2} \)
53 \( 1 - 463. iT - 1.48e5T^{2} \)
59 \( 1 + 580. iT - 2.05e5T^{2} \)
61 \( 1 - 587.T + 2.26e5T^{2} \)
67 \( 1 - 496.T + 3.00e5T^{2} \)
71 \( 1 - 232. iT - 3.57e5T^{2} \)
73 \( 1 - 551. iT - 3.89e5T^{2} \)
79 \( 1 + 437. iT - 4.93e5T^{2} \)
83 \( 1 - 191. iT - 5.71e5T^{2} \)
89 \( 1 + 93.7iT - 7.04e5T^{2} \)
97 \( 1 - 758. iT - 9.12e5T^{2} \)
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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

−9.478787978632632122919258326187, −8.612091182378310763822919043383, −7.84122663269544124071937240036, −6.52075494628358271858184820322, −5.89351905530505041614460209526, −5.35102281301701094872073531924, −4.08939673590923475460355314909, −3.06635917717061385445463986867, −2.28229731918361741728567071751, −0.71992919545207207091354964862, 0.75785331645646641271960170748, 1.89957352341158613670973621929, 2.87414356700936413389106430082, 3.82851574649787662609294687480, 5.35531836889166564674811301865, 5.94547258871327465119035211592, 6.55712249340104884566251668553, 7.48388467477934549739554160299, 8.518052719842867133278329535635, 9.076039931346158572819126173205

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