Properties

Label 2-1344-28.3-c1-0-14
Degree $2$
Conductor $1344$
Sign $0.895 - 0.444i$
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 + (2.5 − 0.866i)7-s + (−0.499 − 0.866i)9-s + (−4.5 − 2.59i)11-s + 6.92i·13-s + 1.73i·15-s + (3 + 1.73i)17-s + (1 + 1.73i)19-s + (−0.500 + 2.59i)21-s + (6 − 3.46i)23-s + (−1 + 1.73i)25-s + 0.999·27-s + 9·29-s + (−0.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (0.670 − 0.387i)5-s + (0.944 − 0.327i)7-s + (−0.166 − 0.288i)9-s + (−1.35 − 0.783i)11-s + 1.92i·13-s + 0.447i·15-s + (0.727 + 0.420i)17-s + (0.229 + 0.397i)19-s + (−0.109 + 0.566i)21-s + (1.25 − 0.722i)23-s + (−0.200 + 0.346i)25-s + 0.192·27-s + 1.67·29-s + (−0.0898 + 0.155i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.895 - 0.444i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.895 - 0.444i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.832860464\)
\(L(\frac12)\) \(\approx\) \(1.832860464\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 0.866i)T \)
good5 \( 1 + (-1.5 + 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.5 + 2.59i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 6.92iT - 13T^{2} \)
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 \( 1 + (-6 + 3.46i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 9T + 29T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 + 3.46iT - 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6 + 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.92iT - 71T^{2} \)
73 \( 1 + (-6 - 3.46i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.5 - 0.866i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 15T + 83T^{2} \)
89 \( 1 + (9 - 5.19i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.66iT - 97T^{2} \)
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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

−9.773581928514216666247761797007, −8.789244199735209798550185760593, −8.296026089908932120632940408190, −7.20292385864110854561293385259, −6.25074602784912804944598357191, −5.24876069722763618331648603604, −4.83374476621401041011151081148, −3.73420425409527926319176198177, −2.38601159394393065722699732028, −1.15830906261202323730076024786, 0.974751033638745402744681020808, 2.39572005934412272789929586921, 3.00381262654390484888141518971, 4.96864073217955286292459699921, 5.24142771320659139165502085924, 6.11712859294788475150300458408, 7.34590468299942910194404687224, 7.76049531249144571311002162546, 8.547760925895572359496019531533, 9.766953831029254469766138321929

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