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Properties

Label 2-1488-372.251-c0-0-1
Degree $2$
Conductor $1488$
Sign $-0.502 + 0.864i$
Analytic cond. $0.742608$
Root an. cond. $0.861747$
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  + (0.104 − 0.994i)3-s + (−0.244 − 1.14i)7-s + (−0.978 − 0.207i)9-s + (0.169 + 0.379i)13-s + (0.604 − 1.35i)19-s + (−1.16 + 0.122i)21-s + (−0.5 − 0.866i)25-s + (−0.309 + 0.951i)27-s + (−0.809 − 0.587i)31-s + (−0.704 + 0.406i)37-s + (0.395 − 0.128i)39-s + (−0.190 − 0.0850i)43-s + (−0.348 + 0.155i)49-s + (−1.28 − 0.743i)57-s + 1.90i·61-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)3-s + (−0.244 − 1.14i)7-s + (−0.978 − 0.207i)9-s + (0.169 + 0.379i)13-s + (0.604 − 1.35i)19-s + (−1.16 + 0.122i)21-s + (−0.5 − 0.866i)25-s + (−0.309 + 0.951i)27-s + (−0.809 − 0.587i)31-s + (−0.704 + 0.406i)37-s + (0.395 − 0.128i)39-s + (−0.190 − 0.0850i)43-s + (−0.348 + 0.155i)49-s + (−1.28 − 0.743i)57-s + 1.90i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.502 + 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1488\)    =    \(2^{4} \cdot 3 \cdot 31\)
Sign: $-0.502 + 0.864i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1488} (623, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1488,\ (\ :0),\ -0.502 + 0.864i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9766960292\)
\(L(\frac12)\) \(\approx\) \(0.9766960292\)
\(L(1)\) 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 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.244 + 1.14i)T + (-0.913 + 0.406i)T^{2} \)
11 \( 1 + (0.104 - 0.994i)T^{2} \)
13 \( 1 + (-0.169 - 0.379i)T + (-0.669 + 0.743i)T^{2} \)
17 \( 1 + (-0.104 - 0.994i)T^{2} \)
19 \( 1 + (-0.604 + 1.35i)T + (-0.669 - 0.743i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.704 - 0.406i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.978 - 0.207i)T^{2} \)
43 \( 1 + (0.190 + 0.0850i)T + (0.669 + 0.743i)T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.913 + 0.406i)T^{2} \)
59 \( 1 + (-0.978 - 0.207i)T^{2} \)
61 \( 1 - 1.90iT - T^{2} \)
67 \( 1 + (-1.64 - 0.951i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.913 + 0.406i)T^{2} \)
73 \( 1 + (-1.47 + 1.33i)T + (0.104 - 0.994i)T^{2} \)
79 \( 1 + (-0.413 + 0.459i)T + (-0.104 - 0.994i)T^{2} \)
83 \( 1 + (0.978 - 0.207i)T^{2} \)
89 \( 1 + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + (-0.604 + 1.86i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.328287373272139780714105408787, −8.537766555744378109883694467518, −7.62106765937741936723963670113, −7.04839286319987380083623864120, −6.43017220077583513118860154987, −5.38379793652504079128515817292, −4.25610020132000682892530790121, −3.26834116472072706325282959127, −2.12372399670474804610864863705, −0.78130977276980580490140170808, 2.00834466136407659445449293584, 3.21994223356112103808607822013, 3.82364614736547032970623540726, 5.27906006013029102590899548573, 5.49220099664410468472059419967, 6.52847776878794947495930382810, 7.82763811950549850547750234049, 8.457906195009926514159031319658, 9.345630772468860954359640781711, 9.725696379404937863842979644768

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