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Properties

Label 2-1476-164.23-c0-0-0
Degree $2$
Conductor $1476$
Sign $0.430 - 0.902i$
Analytic cond. $0.736619$
Root an. cond. $0.858265$
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  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.5 + 0.363i)5-s + (0.809 + 0.587i)8-s + (0.5 + 0.363i)10-s + (−1.11 + 0.363i)13-s + (0.309 − 0.951i)16-s + (−1.11 + 1.53i)17-s + (0.190 − 0.587i)20-s + (−0.190 + 0.587i)25-s + (0.690 + 0.951i)26-s − 32-s + (1.80 + 0.587i)34-s + (−1.30 + 0.951i)37-s − 0.618·40-s + (−0.309 − 0.951i)41-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.809 + 0.587i)4-s + (−0.5 + 0.363i)5-s + (0.809 + 0.587i)8-s + (0.5 + 0.363i)10-s + (−1.11 + 0.363i)13-s + (0.309 − 0.951i)16-s + (−1.11 + 1.53i)17-s + (0.190 − 0.587i)20-s + (−0.190 + 0.587i)25-s + (0.690 + 0.951i)26-s − 32-s + (1.80 + 0.587i)34-s + (−1.30 + 0.951i)37-s − 0.618·40-s + (−0.309 − 0.951i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1476\)    =    \(2^{2} \cdot 3^{2} \cdot 41\)
Sign: $0.430 - 0.902i$
Analytic conductor: \(0.736619\)
Root analytic conductor: \(0.858265\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1476} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1476,\ (\ :0),\ 0.430 - 0.902i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4525586141\)
\(L(\frac12)\) \(\approx\) \(0.4525586141\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 \)
41 \( 1 + (0.309 + 0.951i)T \)
good5 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (-0.809 - 0.587i)T^{2} \)
11 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (0.809 - 0.587i)T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 - 1.61T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
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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

−10.00785526715136254134928780152, −9.041172148044307991864732456556, −8.482131794272686740259074316933, −7.52128306776523829066067741112, −6.85977102001624172645777669088, −5.56175773671396666437935041190, −4.45882581543749184009160549208, −3.81939552269756609021252289208, −2.71722464938743137587005246293, −1.72187692367802884207467527473, 0.39820041942904164398153222382, 2.31449071373157395602683197834, 3.81196484967507742801478338116, 4.85672581199951381243266400031, 5.24337484690584067268427806166, 6.55556378987915228958862767987, 7.14896824646359099122755383828, 7.901241662921398490365849676342, 8.656360233182441269258539476753, 9.397123470279479964855023014786

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