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Properties

Label 2-3200-200.131-c0-0-0
Degree $2$
Conductor $3200$
Sign $0.535 - 0.844i$
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  + (−0.809 − 0.587i)5-s + (−0.309 + 0.951i)9-s + (−1.11 − 0.363i)13-s + (0.5 + 0.363i)17-s + (0.309 + 0.951i)25-s + (1.11 + 1.53i)29-s + (1.80 + 0.587i)37-s + (−0.5 + 1.53i)41-s + (0.809 − 0.587i)45-s + 49-s + (−0.690 − 0.951i)53-s + (−1.11 + 0.363i)61-s + (0.690 + 0.951i)65-s + (0.5 + 1.53i)73-s + (−0.809 − 0.587i)81-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)5-s + (−0.309 + 0.951i)9-s + (−1.11 − 0.363i)13-s + (0.5 + 0.363i)17-s + (0.309 + 0.951i)25-s + (1.11 + 1.53i)29-s + (1.80 + 0.587i)37-s + (−0.5 + 1.53i)41-s + (0.809 − 0.587i)45-s + 49-s + (−0.690 − 0.951i)53-s + (−1.11 + 0.363i)61-s + (0.690 + 0.951i)65-s + (0.5 + 1.53i)73-s + (−0.809 − 0.587i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $0.535 - 0.844i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (831, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ 0.535 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8515938121\)
\(L(\frac12)\) \(\approx\) \(0.8515938121\)
\(L(1)\) 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 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (0.309 - 0.951i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (1.11 + 0.363i)T + (0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.309 - 0.951i)T^{2} \)
37 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
67 \( 1 + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
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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

−8.782427896112830774340402701153, −8.117516060354930706033561435382, −7.68710908942985514055198915558, −6.88212577443652913630883123884, −5.81431839747503610629381558767, −4.90468067476236313860312610992, −4.60731178005967326112313599226, −3.36505382538879753715457722521, −2.56649876536977699830130695600, −1.21149267566684513453863846237, 0.57550376021157352093270267532, 2.37019805331001127032099788618, 3.10297970419880543339263638581, 4.04175712142109640233588516162, 4.69496043857666259922035644227, 5.86967532742020095731713790546, 6.50010902365420439318551557364, 7.36621545267092178493372331617, 7.78578287020536244834661691023, 8.739471376060950900857103786957

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