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Properties

Label 2-3328-8.5-c1-0-20
Degree $2$
Conductor $3328$
Sign $-0.707 - 0.707i$
Analytic cond. $26.5742$
Root an. cond. $5.15501$
Motivic weight $1$
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 + 3i·5-s + 7-s + 2·9-s − 6i·11-s + i·13-s − 3·15-s − 3·17-s + 2i·19-s + i·21-s − 4·25-s + 5i·27-s + 6i·29-s − 4·31-s + 6·33-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.34i·5-s + 0.377·7-s + 0.666·9-s − 1.80i·11-s + 0.277i·13-s − 0.774·15-s − 0.727·17-s + 0.458i·19-s + 0.218i·21-s − 0.800·25-s + 0.962i·27-s + 1.11i·29-s − 0.718·31-s + 1.04·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $-0.707 - 0.707i$
Analytic conductor: \(26.5742\)
Root analytic conductor: \(5.15501\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (1665, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3328,\ (\ :1/2),\ -0.707 - 0.707i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.640554858\)
\(L(\frac12)\) \(\approx\) \(1.640554858\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good3 \( 1 - iT - 3T^{2} \)
5 \( 1 - 3iT - 5T^{2} \)
7 \( 1 - T + 7T^{2} \)
11 \( 1 + 6iT - 11T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 - 2iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 6iT - 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 7iT - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 - 8iT - 61T^{2} \)
67 \( 1 - 14iT - 67T^{2} \)
71 \( 1 - 3T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.866013566446453019821856164960, −8.257471676389210127879094622353, −7.27973223404603266906117069180, −6.70894966424309856169828172450, −5.95845237026512346884389035275, −5.11352457724328466722267212155, −4.07480935273091297767118268393, −3.42212773679443458401107050478, −2.66845648141063194454472935931, −1.35897384063370311495968673190, 0.50480500955533374232175203242, 1.72615396061956506941630155938, 2.17574774195333781676787293138, 3.90027886884733777870311428276, 4.68599449478637230602609381408, 4.98895190193254375161706761511, 6.16101852041468613359726000227, 7.02077708151349494643799282278, 7.62661829176923259084911657122, 8.209553999289665704208192015672

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