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Properties

Label 2-3200-40.37-c0-0-6
Degree $2$
Conductor $3200$
Sign $0.945 + 0.326i$
Analytic cond. $1.59700$
Root an. cond. $1.26372$
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  + (0.707 + 0.707i)3-s − 1.73i·11-s + (−1.22 − 1.22i)17-s + 1.73·19-s + (0.707 − 0.707i)27-s + (1.22 − 1.22i)33-s − 41-s + (1.41 + 1.41i)43-s − i·49-s − 1.73i·51-s + (1.22 + 1.22i)57-s + (−0.707 + 0.707i)67-s + (−1.22 + 1.22i)73-s + 1.00·81-s + (0.707 + 0.707i)83-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)3-s − 1.73i·11-s + (−1.22 − 1.22i)17-s + 1.73·19-s + (0.707 − 0.707i)27-s + (1.22 − 1.22i)33-s − 41-s + (1.41 + 1.41i)43-s − i·49-s − 1.73i·51-s + (1.22 + 1.22i)57-s + (−0.707 + 0.707i)67-s + (−1.22 + 1.22i)73-s + 1.00·81-s + (0.707 + 0.707i)83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.945 + 0.326i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (1857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ 0.945 + 0.326i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.569137843\)
\(L(\frac12)\) \(\approx\) \(1.569137843\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
19 \( 1 - 1.73T + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (0.707 - 0.707i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
89 \( 1 - iT - T^{2} \)
97 \( 1 + iT^{2} \)
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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

−8.879936267194645736622887000700, −8.297944531240731416868352346485, −7.40548711609487964506108697217, −6.56548194757214475071257140886, −5.69434896163578983756070617826, −4.91366455690087993454229859902, −3.97592268608918239929317836153, −3.17249076093474153133820463284, −2.66413080337361737448110774033, −0.925422232863320276145882019951, 1.57461801071528784926108088365, 2.15776361094464177006956170578, 3.15933704244650959081054926518, 4.24739478054425661573697877825, 4.95114622073340629016698419066, 5.97097548463131186166940604380, 6.97722047663466834697828359111, 7.36507716354186926752158555921, 8.009351224585789060974621743149, 8.883089626831967993489974884754

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