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Properties

Label 2-1344-28.3-c1-0-9
Degree $2$
Conductor $1344$
Sign $0.999 - 0.0236i$
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 + (−2.12 + 1.22i)5-s + (−2.63 − 0.272i)7-s + (−0.499 − 0.866i)9-s + (−1.09 − 0.632i)11-s − 2.99i·13-s − 2.45i·15-s + (1.58 + 0.916i)17-s + (−2.07 − 3.60i)19-s + (1.55 − 2.14i)21-s + (5.83 − 3.36i)23-s + (0.507 − 0.879i)25-s + 0.999·27-s + 9.42·29-s + (−4.71 + 8.17i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.499i)3-s + (−0.949 + 0.548i)5-s + (−0.994 − 0.102i)7-s + (−0.166 − 0.288i)9-s + (−0.330 − 0.190i)11-s − 0.831i·13-s − 0.633i·15-s + (0.385 + 0.222i)17-s + (−0.477 − 0.826i)19-s + (0.338 − 0.467i)21-s + (1.21 − 0.702i)23-s + (0.101 − 0.175i)25-s + 0.192·27-s + 1.74·29-s + (−0.847 + 1.46i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.999 - 0.0236i$
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: \((2,\ 1344,\ (\ :1/2),\ 0.999 - 0.0236i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8704611338\)
\(L(\frac12)\) \(\approx\) \(0.8704611338\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.63 + 0.272i)T \)
good5 \( 1 + (2.12 - 1.22i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.09 + 0.632i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.99iT - 13T^{2} \)
17 \( 1 + (-1.58 - 0.916i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.07 + 3.60i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.83 + 3.36i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 9.42T + 29T^{2} \)
31 \( 1 + (4.71 - 8.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.75 - 6.50i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.08iT - 41T^{2} \)
43 \( 1 - 6.27iT - 43T^{2} \)
47 \( 1 + (3.67 + 6.36i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.0358 - 0.0620i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.68 + 2.91i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.61 + 5.55i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.43 + 1.40i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 + (-7.01 - 4.05i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.54 + 0.891i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 5.33T + 83T^{2} \)
89 \( 1 + (-7.42 + 4.28i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.10iT - 97T^{2} \)
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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

−9.795144124501766552811227769621, −8.749767941546117011650655700191, −8.082274842442519356537208215274, −6.95970557479416422205639124865, −6.53239698846771242637503538563, −5.33564792863386612698707735856, −4.48657768514765497591063515053, −3.29142269926349750503193249207, −2.95884435745703627955003100028, −0.57383277588557713080498083332, 0.78788609897453614861270177400, 2.36321210396173223734391396073, 3.59861100399508950088159468972, 4.43141629580583133278726878003, 5.50417181722298386643224792393, 6.38182164178258769109852116088, 7.22192120426360671578447596318, 7.88104351084512096813304999047, 8.777906883343603580003256822965, 9.517591344169661202101690184894

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