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Properties

Label 2-1344-48.35-c1-0-21
Degree $2$
Conductor $1344$
Sign $0.131 - 0.991i$
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  + (1.52 + 0.827i)3-s + (1.23 + 1.23i)5-s + 7-s + (1.62 + 2.51i)9-s + (−3.57 + 3.57i)11-s + (1.58 + 1.58i)13-s + (0.859 + 2.91i)15-s − 2.20i·17-s + (3.76 − 3.76i)19-s + (1.52 + 0.827i)21-s + 3.97i·23-s − 1.92i·25-s + (0.394 + 5.18i)27-s + (−4.75 + 4.75i)29-s − 1.24i·31-s + ⋯
L(s)  = 1  + (0.878 + 0.477i)3-s + (0.554 + 0.554i)5-s + 0.377·7-s + (0.543 + 0.839i)9-s + (−1.07 + 1.07i)11-s + (0.440 + 0.440i)13-s + (0.221 + 0.751i)15-s − 0.535i·17-s + (0.864 − 0.864i)19-s + (0.332 + 0.180i)21-s + 0.828i·23-s − 0.385i·25-s + (0.0759 + 0.997i)27-s + (−0.882 + 0.882i)29-s − 0.223i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.131 - 0.991i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.131 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.495227891\)
\(L(\frac12)\) \(\approx\) \(2.495227891\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.52 - 0.827i)T \)
7 \( 1 - T \)
good5 \( 1 + (-1.23 - 1.23i)T + 5iT^{2} \)
11 \( 1 + (3.57 - 3.57i)T - 11iT^{2} \)
13 \( 1 + (-1.58 - 1.58i)T + 13iT^{2} \)
17 \( 1 + 2.20iT - 17T^{2} \)
19 \( 1 + (-3.76 + 3.76i)T - 19iT^{2} \)
23 \( 1 - 3.97iT - 23T^{2} \)
29 \( 1 + (4.75 - 4.75i)T - 29iT^{2} \)
31 \( 1 + 1.24iT - 31T^{2} \)
37 \( 1 + (6.39 - 6.39i)T - 37iT^{2} \)
41 \( 1 - 3.93T + 41T^{2} \)
43 \( 1 + (1.19 + 1.19i)T + 43iT^{2} \)
47 \( 1 - 8.26T + 47T^{2} \)
53 \( 1 + (2.18 + 2.18i)T + 53iT^{2} \)
59 \( 1 + (-5.45 + 5.45i)T - 59iT^{2} \)
61 \( 1 + (5.10 + 5.10i)T + 61iT^{2} \)
67 \( 1 + (-7.23 + 7.23i)T - 67iT^{2} \)
71 \( 1 + 6.13iT - 71T^{2} \)
73 \( 1 + 10.2iT - 73T^{2} \)
79 \( 1 - 6.53iT - 79T^{2} \)
83 \( 1 + (-12.3 - 12.3i)T + 83iT^{2} \)
89 \( 1 + 9.15T + 89T^{2} \)
97 \( 1 - 7.67T + 97T^{2} \)
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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

−9.603522837346186508349131272343, −9.264709106004604809623411672773, −8.139785426202575248937924523544, −7.43118588636680749805805224873, −6.76622348177573830201200520908, −5.32986250745815277859563519654, −4.81246266941054579185158422631, −3.59281530541698519976028220053, −2.63043835046763156132274533524, −1.81053996761847176532131650306, 0.953880920922736930977927067255, 2.09891251177852868186885904044, 3.15033837825303501192385916825, 4.09794053826590113076615263701, 5.54108687586105574073493017684, 5.85040609500029948720555030420, 7.22948706543160344443447343051, 7.977300285726400751139531278100, 8.539960440702111290032939925320, 9.210054181313213546588431561595

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