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Properties

Label 2-1536-16.13-c1-0-17
Degree $2$
Conductor $1536$
Sign $-0.382 + 0.923i$
Analytic cond. $12.2650$
Root an. cond. $3.50214$
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 + (−2.23 + 2.23i)5-s + 1.74i·7-s + 1.00i·9-s + (−4.57 + 4.57i)11-s + (1 + i)13-s + 3.16·15-s − 2.47·17-s + (−4.57 − 4.57i)19-s + (1.23 − 1.23i)21-s − 5.65i·23-s − 5.00i·25-s + (0.707 − 0.707i)27-s + (4.23 + 4.23i)29-s + 3.90·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.999 + 0.999i)5-s + 0.660i·7-s + 0.333i·9-s + (−1.37 + 1.37i)11-s + (0.277 + 0.277i)13-s + 0.816·15-s − 0.599·17-s + (−1.04 − 1.04i)19-s + (0.269 − 0.269i)21-s − 1.17i·23-s − 1.00i·25-s + (0.136 − 0.136i)27-s + (0.786 + 0.786i)29-s + 0.702·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(12.2650\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1536,\ (\ :1/2),\ -0.382 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1314122370\)
\(L(\frac12)\) \(\approx\) \(0.1314122370\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 0.707i)T \)
good5 \( 1 + (2.23 - 2.23i)T - 5iT^{2} \)
7 \( 1 - 1.74iT - 7T^{2} \)
11 \( 1 + (4.57 - 4.57i)T - 11iT^{2} \)
13 \( 1 + (-1 - i)T + 13iT^{2} \)
17 \( 1 + 2.47T + 17T^{2} \)
19 \( 1 + (4.57 + 4.57i)T + 19iT^{2} \)
23 \( 1 + 5.65iT - 23T^{2} \)
29 \( 1 + (-4.23 - 4.23i)T + 29iT^{2} \)
31 \( 1 - 3.90T + 31T^{2} \)
37 \( 1 + (-7.47 + 7.47i)T - 37iT^{2} \)
41 \( 1 + 2.47iT - 41T^{2} \)
43 \( 1 + (1.08 - 1.08i)T - 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (8.23 - 8.23i)T - 53iT^{2} \)
59 \( 1 + (-6.32 + 6.32i)T - 59iT^{2} \)
61 \( 1 + (1.47 + 1.47i)T + 61iT^{2} \)
67 \( 1 + (8.48 + 8.48i)T + 67iT^{2} \)
71 \( 1 + 9.15iT - 71T^{2} \)
73 \( 1 - 2.94iT - 73T^{2} \)
79 \( 1 + 7.40T + 79T^{2} \)
83 \( 1 + (4.57 + 4.57i)T + 83iT^{2} \)
89 \( 1 - 10iT - 89T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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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

−9.099142346697924440571231009163, −8.241095493522221780996994952445, −7.48617800212671341733195368970, −6.84543602088581748019492948079, −6.18325823704934974949027543028, −4.87630675052785488122526411034, −4.32734651590433532868002033523, −2.78478215279511310620328637541, −2.26498807120745431356380887716, −0.06506538176711891099049079425, 1.01097261390289622769491005105, 2.94731766489674032885058649383, 3.98127353490936479582743363274, 4.57204371470169746172058416851, 5.54113252230950537129475317559, 6.26591692668267521875251533785, 7.55571100749253985564907490127, 8.327565256411910540186106390429, 8.510898490835884243700045578678, 9.904037799688981164121913383429

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