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Properties

Label 2-1344-28.19-c1-0-11
Degree $2$
Conductor $1344$
Sign $0.698 - 0.715i$
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 + (3.33 + 1.92i)5-s + (−1.59 + 2.11i)7-s + (−0.499 + 0.866i)9-s + (1.17 − 0.681i)11-s + 0.369i·13-s − 3.85i·15-s + (3.89 − 2.25i)17-s + (0.0330 − 0.0573i)19-s + (2.62 + 0.323i)21-s + (−2.77 − 1.60i)23-s + (4.93 + 8.54i)25-s + 0.999·27-s + 3.11·29-s + (3.01 + 5.22i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (1.49 + 0.862i)5-s + (−0.602 + 0.798i)7-s + (−0.166 + 0.288i)9-s + (0.355 − 0.205i)11-s + 0.102i·13-s − 0.995i·15-s + (0.945 − 0.545i)17-s + (0.00759 − 0.0131i)19-s + (0.573 + 0.0705i)21-s + (−0.579 − 0.334i)23-s + (0.986 + 1.70i)25-s + 0.192·27-s + 0.579·29-s + (0.542 + 0.939i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.698 - 0.715i$
Analytic conductor: \(10.7318\)
Root analytic conductor: \(3.27595\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (1279, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :1/2),\ 0.698 - 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.862176113\)
\(L(\frac12)\) \(\approx\) \(1.862176113\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 2.11i)T \)
good5 \( 1 + (-3.33 - 1.92i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.17 + 0.681i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.369iT - 13T^{2} \)
17 \( 1 + (-3.89 + 2.25i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0330 + 0.0573i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.77 + 1.60i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.11T + 29T^{2} \)
31 \( 1 + (-3.01 - 5.22i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.74 - 4.75i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8.45iT - 41T^{2} \)
43 \( 1 - 6.30iT - 43T^{2} \)
47 \( 1 + (-0.712 + 1.23i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.27 + 2.20i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.71 - 2.97i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.23 + 0.715i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-8.45 + 4.88i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 + (1.56 - 0.900i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.8 + 6.24i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 + (-1.11 - 0.646i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.88iT - 97T^{2} \)
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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

−9.908755153564697264365865760380, −9.042589690446816454810891824867, −8.118459137708753769806051057517, −6.90226871416883063767137564506, −6.39657014904753304590237730460, −5.81036299534247442203133843915, −4.96741387163344002271345101792, −3.19992885630399411689697146216, −2.56015085855146528549165359988, −1.41517597615019151410395243076, 0.860724215737522171079632256465, 2.08324948047265111741992310647, 3.54815978398820083207255621740, 4.43177403738266973324893438217, 5.47040809306111250574463379169, 5.98071807740791172228316349475, 6.85878088436389222487602666242, 7.989888232300466589172033254365, 9.019085323993206369917827288578, 9.599452504557505134409777743124

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