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Properties

Label 2-1664-104.21-c0-0-1
Degree $2$
Conductor $1664$
Sign $0.289 - 0.957i$
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  + i·3-s + (0.707 − 0.707i)5-s + (−0.707 + 0.707i)7-s + (1 + i)11-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)15-s − i·17-s + (−0.707 − 0.707i)21-s − 1.41i·23-s + i·27-s + 1.41i·29-s + (−1 + i)33-s + 1.00i·35-s + (0.707 + 0.707i)37-s + (−0.707 − 0.707i)39-s + ⋯
L(s)  = 1  + i·3-s + (0.707 − 0.707i)5-s + (−0.707 + 0.707i)7-s + (1 + i)11-s + (−0.707 + 0.707i)13-s + (0.707 + 0.707i)15-s − i·17-s + (−0.707 − 0.707i)21-s − 1.41i·23-s + i·27-s + 1.41i·29-s + (−1 + i)33-s + 1.00i·35-s + (0.707 + 0.707i)37-s + (−0.707 − 0.707i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.289 - 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1664\)    =    \(2^{7} \cdot 13\)
Sign: $0.289 - 0.957i$
Analytic conductor: \(0.830444\)
Root analytic conductor: \(0.911287\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1664} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1664,\ (\ :0),\ 0.289 - 0.957i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.229461812\)
\(L(\frac12)\) \(\approx\) \(1.229461812\)
\(L(1)\) 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 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 - iT - T^{2} \)
5 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
7 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
11 \( 1 + (-1 - i)T + iT^{2} \)
17 \( 1 + iT - T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
53 \( 1 + 1.41iT - T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 + (1 - i)T - iT^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 + (1 + i)T + iT^{2} \)
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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

−9.685878329027556008283051895858, −9.193457264687107990085595210004, −8.575958578868921461517541830260, −7.02168128847209343022415076357, −6.62284203135571522578207491089, −5.33627878124151958675192202679, −4.82237243320119109399994840625, −4.04790747763811864067564084750, −2.81331842511282801926428082819, −1.68121218051828596162805361813, 1.04894820454447513609737856557, 2.24505963915971078587175628493, 3.30725232921526345758596080282, 4.17808876647131256465966920375, 5.89263123652349164171185611338, 6.11386881904091137424480576108, 7.01417342748466124869809461681, 7.57780966302888720172622069584, 8.461840380668945344717631162386, 9.649467301072427436392084721694

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