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Properties

Label 2-1344-28.3-c1-0-2
Degree $2$
Conductor $1344$
Sign $-0.311 - 0.950i$
Analytic cond. $10.7318$
Root an. cond. $3.27595$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)3-s + (−1.5 + 0.866i)5-s + (0.5 + 2.59i)7-s + (−0.499 − 0.866i)9-s + (−1.5 − 0.866i)11-s + 1.73i·15-s + (3 + 1.73i)17-s + (−1 − 1.73i)19-s + (2.5 + 0.866i)21-s + (−1 + 1.73i)25-s − 0.999·27-s − 9·29-s + (−2.5 + 4.33i)31-s + (−1.5 + 0.866i)33-s + (−3 − 3.46i)35-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−0.670 + 0.387i)5-s + (0.188 + 0.981i)7-s + (−0.166 − 0.288i)9-s + (−0.452 − 0.261i)11-s + 0.447i·15-s + (0.727 + 0.420i)17-s + (−0.229 − 0.397i)19-s + (0.545 + 0.188i)21-s + (−0.200 + 0.346i)25-s − 0.192·27-s − 1.67·29-s + (−0.449 + 0.777i)31-s + (−0.261 + 0.150i)33-s + (−0.507 − 0.585i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.311 - 0.950i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.311 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9422829347\)
\(L(\frac12)\) \(\approx\) \(0.9422829347\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 2.59i)T \)
good5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.5 + 0.866i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + (-3 - 1.73i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 + 1.73i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (2.5 - 4.33i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5 - 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.5 - 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-12 - 6.92i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + (6 + 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 + 2.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 3T + 83T^{2} \)
89 \( 1 + (3 - 1.73i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 19.0iT - 97T^{2} \)
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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

−9.711788925346536340007909994464, −8.941621305503355469216029597254, −8.054529692406134929759072928636, −7.66816594429029605797847809387, −6.60065830135402369774172717899, −5.78455460248271172821949467868, −4.86816728449235627105783135258, −3.54065892163771220558503280851, −2.81741687777095973756949981560, −1.59461849054744724376829020769, 0.36640804972036081150218055306, 2.04160976586617923692706113014, 3.52941122990749073035678738078, 4.07874977413171575318447688118, 4.98074917355892730828290375723, 5.90105322233382219591464112123, 7.32299482226001572799684607846, 7.67612785663974542941572876259, 8.475157973547118793149750402638, 9.516433493159246371159951381456

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