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Properties

Label 2-1512-168.53-c0-0-1
Degree $2$
Conductor $1512$
Sign $-0.832 - 0.553i$
Analytic cond. $0.754586$
Root an. cond. $0.868669$
Motivic weight $0$
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)2-s + (−0.499 + 0.866i)4-s + (1 + 1.73i)5-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.999 + 1.73i)10-s + (−0.5 + 0.866i)11-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 1.99·20-s − 0.999·22-s + (−1.49 + 2.59i)25-s + 0.999·28-s + 29-s + (0.5 − 0.866i)31-s + (0.499 − 0.866i)32-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1 + 1.73i)5-s + (−0.5 − 0.866i)7-s − 0.999·8-s + (−0.999 + 1.73i)10-s + (−0.5 + 0.866i)11-s + (0.499 − 0.866i)14-s + (−0.5 − 0.866i)16-s − 1.99·20-s − 0.999·22-s + (−1.49 + 2.59i)25-s + 0.999·28-s + 29-s + (0.5 − 0.866i)31-s + (0.499 − 0.866i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.832 - 0.553i$
Analytic conductor: \(0.754586\)
Root analytic conductor: \(0.868669\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :0),\ -0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.382601546\)
\(L(\frac12)\) \(\approx\) \(1.382601546\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (0.5 + 0.866i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T + T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−9.988100436282609546683849615854, −9.378366899368923992853388883516, −8.009980884085348127152751471903, −7.27421347621101733309611860769, −6.72505517689478139657666045799, −6.18093034966615582899759535044, −5.21505584794090711478529307604, −4.08592782930403983622436601918, −3.14857155654038918398815354666, −2.32067434408937287447326352935, 0.974220772368456107092674549360, 2.15421677635689605844984170143, 3.08170338825545638706794324898, 4.42069401590294545422441860899, 5.18650165504060676898844361185, 5.74709766774326415217261275120, 6.40403060421369133313616565384, 8.296773198397044591400360927913, 8.730894382571811002654354041195, 9.408640769698783132152085830997

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