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Properties

Label 2-1344-56.27-c3-0-9
Degree $2$
Conductor $1344$
Sign $-0.924 + 0.382i$
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 + 4.35·5-s + (−7.09 + 17.1i)7-s − 9·9-s + 43.1·11-s − 26.5·13-s + 13.0i·15-s − 76.6i·17-s + 58.7i·19-s + (−51.3 − 21.2i)21-s + 0.747i·23-s − 106.·25-s − 27i·27-s + 64.9i·29-s − 165.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 0.389·5-s + (−0.382 + 0.923i)7-s − 0.333·9-s + 1.18·11-s − 0.565·13-s + 0.224i·15-s − 1.09i·17-s + 0.709i·19-s + (−0.533 − 0.221i)21-s + 0.00677i·23-s − 0.848·25-s − 0.192i·27-s + 0.416i·29-s − 0.958·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.382i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.924 + 0.382i$
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: \((2,\ 1344,\ (\ :3/2),\ -0.924 + 0.382i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6204693262\)
\(L(\frac12)\) \(\approx\) \(0.6204693262\)
\(L(\frac{5}{2})\) 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 \( 1 + (7.09 - 17.1i)T \)
good5 \( 1 - 4.35T + 125T^{2} \)
11 \( 1 - 43.1T + 1.33e3T^{2} \)
13 \( 1 + 26.5T + 2.19e3T^{2} \)
17 \( 1 + 76.6iT - 4.91e3T^{2} \)
19 \( 1 - 58.7iT - 6.85e3T^{2} \)
23 \( 1 - 0.747iT - 1.21e4T^{2} \)
29 \( 1 - 64.9iT - 2.43e4T^{2} \)
31 \( 1 + 165.T + 2.97e4T^{2} \)
37 \( 1 - 389. iT - 5.06e4T^{2} \)
41 \( 1 - 290. iT - 6.89e4T^{2} \)
43 \( 1 + 32.4T + 7.95e4T^{2} \)
47 \( 1 - 368.T + 1.03e5T^{2} \)
53 \( 1 + 588. iT - 1.48e5T^{2} \)
59 \( 1 + 471. iT - 2.05e5T^{2} \)
61 \( 1 + 448.T + 2.26e5T^{2} \)
67 \( 1 + 707.T + 3.00e5T^{2} \)
71 \( 1 - 997. iT - 3.57e5T^{2} \)
73 \( 1 - 1.22e3iT - 3.89e5T^{2} \)
79 \( 1 + 160. iT - 4.93e5T^{2} \)
83 \( 1 + 1.00e3iT - 5.71e5T^{2} \)
89 \( 1 + 1.13e3iT - 7.04e5T^{2} \)
97 \( 1 + 729. iT - 9.12e5T^{2} \)
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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

−9.660408985868655653856066051097, −9.095821295382743222311803736630, −8.295210177711129376138101504257, −7.14962221245948768787387606835, −6.28562681097550683478671039490, −5.54035548489580269713149432699, −4.69708389504299102771144008667, −3.60842929632039784903002234711, −2.70294446055440889546722310691, −1.55286097163037877396245741936, 0.13772510604666323830146097929, 1.30712925224877013918746027362, 2.28453069572297189699462772545, 3.64390577382954093760828066509, 4.31663194260821686667287424791, 5.70047545203250104581500153412, 6.33315307308896155435486959366, 7.22002693610198290407406122431, 7.70274754073219019835045327896, 9.028570639992148108129251236379

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