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Properties

Label 2-1344-112.27-c1-0-28
Degree $2$
Conductor $1344$
Sign $-0.414 + 0.909i$
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  + (0.707 − 0.707i)3-s + (1.07 − 1.07i)5-s + (−0.929 − 2.47i)7-s − 1.00i·9-s + (1.12 − 1.12i)11-s + (0.380 + 0.380i)13-s − 1.51i·15-s − 4.93i·17-s + (0.00735 − 0.00735i)19-s + (−2.40 − 1.09i)21-s − 1.62·23-s + 2.70i·25-s + (−0.707 − 0.707i)27-s + (−3.69 + 3.69i)29-s + 1.21·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.479 − 0.479i)5-s + (−0.351 − 0.936i)7-s − 0.333i·9-s + (0.338 − 0.338i)11-s + (0.105 + 0.105i)13-s − 0.391i·15-s − 1.19i·17-s + (0.00168 − 0.00168i)19-s + (−0.525 − 0.238i)21-s − 0.338·23-s + 0.540i·25-s + (−0.136 − 0.136i)27-s + (−0.686 + 0.686i)29-s + 0.217·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.414 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.414 + 0.909i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1231, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ -0.414 + 0.909i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.802613614\)
\(L(\frac12)\) \(\approx\) \(1.802613614\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 + 0.707i)T \)
7 \( 1 + (0.929 + 2.47i)T \)
good5 \( 1 + (-1.07 + 1.07i)T - 5iT^{2} \)
11 \( 1 + (-1.12 + 1.12i)T - 11iT^{2} \)
13 \( 1 + (-0.380 - 0.380i)T + 13iT^{2} \)
17 \( 1 + 4.93iT - 17T^{2} \)
19 \( 1 + (-0.00735 + 0.00735i)T - 19iT^{2} \)
23 \( 1 + 1.62T + 23T^{2} \)
29 \( 1 + (3.69 - 3.69i)T - 29iT^{2} \)
31 \( 1 - 1.21T + 31T^{2} \)
37 \( 1 + (6.70 + 6.70i)T + 37iT^{2} \)
41 \( 1 - 2.08T + 41T^{2} \)
43 \( 1 + (-7.89 + 7.89i)T - 43iT^{2} \)
47 \( 1 - 2.09T + 47T^{2} \)
53 \( 1 + (5.28 + 5.28i)T + 53iT^{2} \)
59 \( 1 + (-9.39 - 9.39i)T + 59iT^{2} \)
61 \( 1 + (2.45 + 2.45i)T + 61iT^{2} \)
67 \( 1 + (4.70 + 4.70i)T + 67iT^{2} \)
71 \( 1 + 5.16T + 71T^{2} \)
73 \( 1 + 7.83T + 73T^{2} \)
79 \( 1 - 4.65iT - 79T^{2} \)
83 \( 1 + (-0.694 + 0.694i)T - 83iT^{2} \)
89 \( 1 - 16.3T + 89T^{2} \)
97 \( 1 - 15.8iT - 97T^{2} \)
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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

−9.182420709700014726949000041922, −8.787821940255488736494951494972, −7.48611183247941046236168510569, −7.14678086229632764511245381473, −6.08141135218517231464495461894, −5.19943548832458047996988989353, −4.06525592472356727524548333487, −3.20109182293399308391512770529, −1.89274513515373751519536338012, −0.69542427856400611052073328532, 1.83615263389835258230458477782, 2.73848276224476436790978064190, 3.73143518579112187559081899198, 4.74575982121856913583852884838, 5.96108789382424892974131438965, 6.30088054577222889338102486908, 7.52867133405733059586576529560, 8.422547975443612791491113373854, 9.088573612158249258701408614540, 9.898335800080776813975160009136

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