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Properties

Label 2-2028-39.29-c0-0-0
Degree $2$
Conductor $2028$
Sign $-0.664 - 0.746i$
Analytic cond. $1.01210$
Root an. cond. $1.00603$
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)3-s + (−0.866 + 1.5i)7-s + (−0.499 + 0.866i)9-s − 1.73·21-s + 25-s − 0.999·27-s − 1.73·31-s + (−0.5 + 0.866i)43-s + (−1 − 1.73i)49-s + (−0.5 + 0.866i)61-s + (−0.866 − 1.49i)63-s + (0.866 + 1.5i)67-s + 1.73·73-s + (0.5 + 0.866i)75-s + 79-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.866 + 1.5i)7-s + (−0.499 + 0.866i)9-s − 1.73·21-s + 25-s − 0.999·27-s − 1.73·31-s + (−0.5 + 0.866i)43-s + (−1 − 1.73i)49-s + (−0.5 + 0.866i)61-s + (−0.866 − 1.49i)63-s + (0.866 + 1.5i)67-s + 1.73·73-s + (0.5 + 0.866i)75-s + 79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2028\)    =    \(2^{2} \cdot 3 \cdot 13^{2}\)
Sign: $-0.664 - 0.746i$
Analytic conductor: \(1.01210\)
Root analytic conductor: \(1.00603\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2028} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2028,\ (\ :0),\ -0.664 - 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.065995783\)
\(L(\frac12)\) \(\approx\) \(1.065995783\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good5 \( 1 - T^{2} \)
7 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + 1.73T + T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 - 1.73T + T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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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

−9.477573250544086423753903097817, −8.928737803267104914557559754706, −8.409141620134030306329376521581, −7.34688426580669827307255785849, −6.31050184442944251618123836383, −5.54175481087427930192594200113, −4.86340897215149149484346394028, −3.67658124194440964129395766014, −2.95704701128708717350004676892, −2.11482243059191104292384473397, 0.69874770320213459157993883171, 1.98849666271633357621488127018, 3.30345605606611983111480920873, 3.75585220173567784397022089098, 5.00700659218045369388411140485, 6.21112043807360551317695804149, 6.85462729702733099696646125987, 7.37575455864994750061719247480, 8.113620231862634430114826485232, 9.084712958643671111673474860659

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