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Properties

Label 2-1332-148.147-c0-0-0
Degree $2$
Conductor $1332$
Sign $-i$
Analytic cond. $0.664754$
Root an. cond. $0.815324$
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)2-s + 1.00i·4-s + 1.41i·5-s + 2i·7-s + (0.707 − 0.707i)8-s + (1.00 − 1.00i)10-s + (1.41 − 1.41i)14-s − 1.00·16-s − 1.41i·17-s − 1.41·20-s − 1.41·23-s − 1.00·25-s − 2.00·28-s + 1.41i·29-s + (0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + 1.41i·5-s + 2i·7-s + (0.707 − 0.707i)8-s + (1.00 − 1.00i)10-s + (1.41 − 1.41i)14-s − 1.00·16-s − 1.41i·17-s − 1.41·20-s − 1.41·23-s − 1.00·25-s − 2.00·28-s + 1.41i·29-s + (0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1332\)    =    \(2^{2} \cdot 3^{2} \cdot 37\)
Sign: $-i$
Analytic conductor: \(0.664754\)
Root analytic conductor: \(0.815324\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1332} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1332,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6602424508\)
\(L(\frac12)\) \(\approx\) \(0.6602424508\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−9.892056536591717787071047078648, −9.371386255470186008935881934893, −8.571377289541421886647726263158, −7.74331922563593644290663017934, −6.85386042430630659094192996300, −6.05324551244972753635695859349, −4.97076956644728292778138433325, −3.46361225721334591699090925779, −2.70425552170417996162447736291, −2.10248082468340332890912845936, 0.71403658330381082758101991000, 1.72900014117092613444099523696, 4.10672817019103064091094510561, 4.35000340292243973979457524416, 5.60515086101696016820708607005, 6.39576061096915889048739935070, 7.40190470981591793843066613870, 8.081127869578809371764089097926, 8.533047380232175134344017996767, 9.758375538032464746146122612807

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