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Properties

Label 2-1344-28.3-c1-0-4
Degree $2$
Conductor $1344$
Sign $-0.0537 - 0.998i$
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 + (−2.33 + 1.35i)5-s + (2.63 + 0.192i)7-s + (−0.499 − 0.866i)9-s + (−1.22 − 0.708i)11-s + 5.69i·13-s + 2.70i·15-s + (−1.69 − 0.979i)17-s + (0.361 + 0.626i)19-s + (1.48 − 2.18i)21-s + (−0.562 + 0.324i)23-s + (1.14 − 1.98i)25-s − 0.999·27-s − 6.06·29-s + (−2.49 + 4.32i)31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (−1.04 + 0.603i)5-s + (0.997 + 0.0728i)7-s + (−0.166 − 0.288i)9-s + (−0.370 − 0.213i)11-s + 1.57i·13-s + 0.697i·15-s + (−0.411 − 0.237i)17-s + (0.0829 + 0.143i)19-s + (0.324 − 0.477i)21-s + (−0.117 + 0.0677i)23-s + (0.229 − 0.396i)25-s − 0.192·27-s − 1.12·29-s + (−0.448 + 0.776i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0537 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0537 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $-0.0537 - 0.998i$
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: \((2,\ 1344,\ (\ :1/2),\ -0.0537 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.098690167\)
\(L(\frac12)\) \(\approx\) \(1.098690167\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 - 0.192i)T \)
good5 \( 1 + (2.33 - 1.35i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.22 + 0.708i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 5.69iT - 13T^{2} \)
17 \( 1 + (1.69 + 0.979i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.361 - 0.626i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.562 - 0.324i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 6.06T + 29T^{2} \)
31 \( 1 + (2.49 - 4.32i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.576 - 0.999i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.58iT - 41T^{2} \)
43 \( 1 - 10.8iT - 43T^{2} \)
47 \( 1 + (-4.81 - 8.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.26 + 10.8i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (0.434 - 0.751i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (11.7 - 6.80i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.0459 + 0.0265i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.0iT - 71T^{2} \)
73 \( 1 + (-13.5 - 7.82i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (9.95 - 5.74i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 2.95T + 83T^{2} \)
89 \( 1 + (7.13 - 4.12i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.01iT - 97T^{2} \)
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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

−9.694620863073044322460073816095, −8.808319647716174414688903523082, −8.133437952617189657496330088727, −7.34625834251694885643697989259, −6.89344378447830889086781034006, −5.71669638143536737205564308748, −4.54908267160955916282694096032, −3.81662918370012088677039974895, −2.63427262833979598584464224035, −1.53513968201487218989270805083, 0.43796790052605194911055614925, 2.12438754711012046776090496212, 3.45933291532086314975606900744, 4.25476749093509636510572219438, 5.05694346499466058503731186117, 5.77589417795518405440053606015, 7.46431357637557265890042198172, 7.77663189155397983015080939244, 8.536021867856168851471193258615, 9.202336194334092400511443207929

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