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Properties

Label 2-3200-8.3-c0-0-7
Degree $2$
Conductor $3200$
Sign $-0.707 + 0.707i$
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  − 1.41·3-s − 1.41i·7-s + 1.00·9-s + 2.00i·21-s − 1.41i·23-s − 1.41·43-s + 1.41i·47-s − 1.00·49-s − 2i·61-s − 1.41i·63-s − 1.41·67-s + 2.00i·69-s − 0.999·81-s + 1.41·83-s − 2·89-s + ⋯
L(s)  = 1  − 1.41·3-s − 1.41i·7-s + 1.00·9-s + 2.00i·21-s − 1.41i·23-s − 1.41·43-s + 1.41i·47-s − 1.00·49-s − 2i·61-s − 1.41i·63-s − 1.41·67-s + 2.00i·69-s − 0.999·81-s + 1.41·83-s − 2·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (2751, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4662291861\)
\(L(\frac12)\) \(\approx\) \(0.4662291861\)
\(L(1)\) 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 \( 1 + 1.41T + T^{2} \)
7 \( 1 + 1.41iT - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.41T + T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + 2iT - T^{2} \)
67 \( 1 + 1.41T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.41T + T^{2} \)
89 \( 1 + 2T + T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.434980925000736924082381756591, −7.67946998145874202635569305191, −6.78088575183960617080879140206, −6.47874590692073622423268580621, −5.52732331744100619750104701623, −4.67845733796571483895045468403, −4.19001649200167457050129959681, −3.03760158712363483108825850329, −1.48330854408455664632548220921, −0.36335909161746959152283264256, 1.44163976819106055781135749206, 2.58631054319044862484356399364, 3.70129910683098286515361163689, 4.89372217625671213242701713134, 5.41747848359478248033828695925, 5.94702206326905230306988255712, 6.66361190481337638512762995721, 7.50013373852945488180917577972, 8.488754076221090544177108265801, 9.115232391732079960842745890018

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