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Properties

Label 2-1664-104.85-c0-0-0
Degree $2$
Conductor $1664$
Sign $-0.852 + 0.522i$
Analytic cond. $0.830444$
Root an. cond. $0.911287$
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  + (−1.36 + 1.36i)5-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)13-s + (−0.866 + 0.5i)17-s − 2.73i·25-s + (−0.866 − 0.5i)29-s + (0.5 − 0.133i)37-s + (−1.86 + 0.5i)41-s + (1.86 + 0.499i)45-s + (−0.866 − 0.5i)49-s − 1.73i·53-s + (−1.5 + 0.866i)61-s + (1.86 − 0.499i)65-s + (1.36 − 1.36i)73-s + (−0.499 + 0.866i)81-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)5-s + (−0.5 − 0.866i)9-s + (−0.866 − 0.5i)13-s + (−0.866 + 0.5i)17-s − 2.73i·25-s + (−0.866 − 0.5i)29-s + (0.5 − 0.133i)37-s + (−1.86 + 0.5i)41-s + (1.86 + 0.499i)45-s + (−0.866 − 0.5i)49-s − 1.73i·53-s + (−1.5 + 0.866i)61-s + (1.86 − 0.499i)65-s + (1.36 − 1.36i)73-s + (−0.499 + 0.866i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1664\)    =    \(2^{7} \cdot 13\)
Sign: $-0.852 + 0.522i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1664} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1664,\ (\ :0),\ -0.852 + 0.522i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.03976340149\)
\(L(\frac12)\) \(\approx\) \(0.03976340149\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 + (0.866 + 0.5i)T \)
good3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (1.36 - 1.36i)T - iT^{2} \)
7 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
41 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + 1.73iT - T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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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

−9.228409824813959764837833626803, −8.196598641791591058737723024407, −7.71004497571216406348848847030, −6.74068988953393812461827396520, −6.36155937326452825372859151304, −5.02718901224589264093661952642, −3.90149401836746929931536941871, −3.32976663167496968125966879257, −2.38463557701641958282149550573, −0.02908911747078765163737097158, 1.74752263547627394428401010209, 3.11253171366067046380573896329, 4.32173653278857551919963377261, 4.78735313925148670847066592521, 5.53727845418375049998595239592, 6.95899687281277291026414748150, 7.63088109770767674509761874160, 8.315761603218261739950486559859, 8.953890238485337355403477827879, 9.627160228462341842120469354525

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