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Properties

Label 2-1344-7.2-c1-0-20
Degree $2$
Conductor $1344$
Sign $0.386 + 0.922i$
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 − 2.59i)5-s + (−0.5 + 2.59i)7-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + 4·13-s − 3·15-s + (−2 − 3.46i)17-s + (2.5 − 0.866i)21-s + (4 − 6.92i)23-s + (−2 − 3.46i)25-s + 0.999·27-s + 7·29-s + (5.5 + 9.52i)31-s + (0.499 − 0.866i)33-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.670 − 1.16i)5-s + (−0.188 + 0.981i)7-s + (−0.166 + 0.288i)9-s + (0.150 + 0.261i)11-s + 1.10·13-s − 0.774·15-s + (−0.485 − 0.840i)17-s + (0.545 − 0.188i)21-s + (0.834 − 1.44i)23-s + (−0.400 − 0.692i)25-s + 0.192·27-s + 1.29·29-s + (0.987 + 1.71i)31-s + (0.0870 − 0.150i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.734697406\)
\(L(\frac12)\) \(\approx\) \(1.734697406\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7T + 29T^{2} \)
31 \( 1 + (-5.5 - 9.52i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (1 - 1.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.5 + 9.52i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.5 + 6.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 + 8.66i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-3 - 5.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−9.186874160606983069984929416596, −8.739257275064680281031669429951, −8.129923352657131114395774051163, −6.57117942376118690062142446769, −6.37990816668717623083536430168, −5.06163116513150962783386038976, −4.84034314503276473901531131539, −3.10046686024821182551724976966, −1.98184731264477549097822052644, −0.873195450882738156505419366204, 1.26248597761430183616232730659, 2.83685834219060416299373919959, 3.66771491238042644943726516818, 4.53249032800872432696424561415, 5.92032540992724889093630859078, 6.32655495123957651176649343127, 7.13039351053344942575014776823, 8.143548307247605719678740202827, 9.125591191316717170756914902042, 10.06403949339596612909995604869

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