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Properties

Label 2-1332-148.147-c0-0-2
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

\(L(\frac{1}{2})\) \(\approx\) \(1.497182289\)
\(L(\frac12)\) \(\approx\) \(1.497182289\)
\(L(1)\) 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

−9.536514021284968627549872751522, −8.960838945991013022801378775495, −8.414778554632027298497333622971, −7.80162547623979677858169270845, −6.31494445234611737460733423633, −5.83519610973586499523029156412, −5.06465265240301241344898402296, −4.39234271263682380976384369493, −3.07719660515469963840583617553, −1.93885816684015931478909345699, 1.11343943498100243464676815912, 2.77612591838872342442985791975, 3.36725699574225717923528342956, 4.30956838860535088509575402938, 5.16834423345759994140862797861, 6.54000686327996443523670903605, 7.00585498757766112714450620749, 7.56839259189705911656198883057, 9.188191971362504663544386408686, 9.989054943683312493424151725686

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