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Properties

Label 2-3332-476.135-c0-0-15
Degree $2$
Conductor $3332$
Sign $0.605 - 0.795i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
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  + (0.5 − 0.866i)2-s + (−0.866 − 1.5i)3-s + (−0.499 − 0.866i)4-s − 1.73·6-s − 0.999·8-s + (−1 + 1.73i)9-s + (−0.866 − 1.5i)11-s + (−0.866 + 1.49i)12-s − 13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (1 + 1.73i)18-s − 1.73·22-s + (0.866 + 1.49i)24-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.866 − 1.5i)3-s + (−0.499 − 0.866i)4-s − 1.73·6-s − 0.999·8-s + (−1 + 1.73i)9-s + (−0.866 − 1.5i)11-s + (−0.866 + 1.49i)12-s − 13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (1 + 1.73i)18-s − 1.73·22-s + (0.866 + 1.49i)24-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (2039, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3439922494\)
\(L(\frac12)\) \(\approx\) \(0.3439922494\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
17 \( 1 + (-0.5 - 0.866i)T \)
good3 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + T + T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + 1.73T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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

−8.116556080658556640189013315974, −7.38643512297835426813599556757, −6.41211125030542698911522688627, −5.77427487029256289685927859270, −5.40841349069562449101373985487, −4.32747359206574185109780906838, −3.07380540530113312807222696984, −2.34984408453994711737628692473, −1.31824153352001322417227971608, −0.19625472632838879179832752607, 2.52473550059152230458130804148, 3.53857309532607205114071896618, 4.41065709988516126275875600245, 5.03876331686918251609013498655, 5.29216042312818692942098105083, 6.23831461657749057877131046693, 7.27505930644004895212895044547, 7.60591989301034614780207041384, 8.831895358681502998728009516566, 9.626334997842571290574074949998

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