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Properties

Label 2-1536-24.5-c0-0-0
Degree $2$
Conductor $1536$
Sign $-0.707 - 0.707i$
Analytic cond. $0.766563$
Root an. cond. $0.875536$
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  + (−0.707 + 0.707i)3-s − 1.00i·9-s − 1.41·11-s + 2i·17-s + 1.41i·19-s − 25-s + (0.707 + 0.707i)27-s + (1.00 − 1.00i)33-s + 1.41i·43-s − 49-s + (−1.41 − 1.41i)51-s + (−1.00 − 1.00i)57-s − 1.41·59-s − 1.41i·67-s + (0.707 − 0.707i)75-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)3-s − 1.00i·9-s − 1.41·11-s + 2i·17-s + 1.41i·19-s − 25-s + (0.707 + 0.707i)27-s + (1.00 − 1.00i)33-s + 1.41i·43-s − 49-s + (−1.41 − 1.41i)51-s + (−1.00 − 1.00i)57-s − 1.41·59-s − 1.41i·67-s + (0.707 − 0.707i)75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(0.766563\)
Root analytic conductor: \(0.875536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1536,\ (\ :0),\ -0.707 - 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5327565696\)
\(L(\frac12)\) \(\approx\) \(0.5327565696\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.707 - 0.707i)T \)
good5 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - 2iT - T^{2} \)
19 \( 1 - 1.41iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.41iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 1.41iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 - 2iT - T^{2} \)
97 \( 1 + T^{2} \)
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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

−10.12872591559594378665618797872, −9.372857366841205592021173125009, −8.165291315313074713755228204170, −7.82797390122473672431861055716, −6.34261590569526155757475507431, −5.90420745528580941493201554281, −5.03409068897933160325336658892, −4.08979718002394968940076734233, −3.27164727409960944429566168107, −1.75546975394448676492689802751, 0.44398246172531131990155054610, 2.17957965378170129713014414045, 2.99122889970591481800301695076, 4.70316113257124598949347237997, 5.17805838192197008278135892211, 6.05491970730050404981486930720, 7.17232937495162432170198265853, 7.43552023842044001538601113732, 8.420196418201900786109670975017, 9.403711803286296275258981305793

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