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Properties

Label 2-3200-200.59-c0-0-0
Degree $2$
Conductor $3200$
Sign $-0.728 - 0.684i$
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  + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)9-s + (−0.5 + 0.363i)13-s + (1.11 + 0.363i)17-s + (−0.809 − 0.587i)25-s + (−1.11 + 0.363i)29-s + (−1.30 + 0.951i)37-s + (0.5 − 0.363i)41-s + (−0.309 − 0.951i)45-s − 49-s + (0.190 + 0.587i)53-s + (−1.11 + 1.53i)61-s + (−0.190 − 0.587i)65-s + (−1.11 + 1.53i)73-s + (0.309 − 0.951i)81-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)9-s + (−0.5 + 0.363i)13-s + (1.11 + 0.363i)17-s + (−0.809 − 0.587i)25-s + (−1.11 + 0.363i)29-s + (−1.30 + 0.951i)37-s + (0.5 − 0.363i)41-s + (−0.309 − 0.951i)45-s − 49-s + (0.190 + 0.587i)53-s + (−1.11 + 1.53i)61-s + (−0.190 − 0.587i)65-s + (−1.11 + 1.53i)73-s + (0.309 − 0.951i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3200\)    =    \(2^{7} \cdot 5^{2}\)
Sign: $-0.728 - 0.684i$
Analytic conductor: \(1.59700\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3200} (959, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3200,\ (\ :0),\ -0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7165451209\)
\(L(\frac12)\) \(\approx\) \(0.7165451209\)
\(L(1)\) 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 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.809 - 0.587i)T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + (0.309 + 0.951i)T^{2} \)
13 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T^{2} \)
23 \( 1 + (0.309 + 0.951i)T^{2} \)
29 \( 1 + (1.11 - 0.363i)T + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.809 + 0.587i)T^{2} \)
71 \( 1 + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
97 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
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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

−9.053591843438523847925072198508, −8.306472543612619701382175460215, −7.54174847209871883737635291014, −7.07454070782375986737002385243, −6.04255598830552611057907101856, −5.47426917507994459284615259949, −4.46600258136884012898375722130, −3.44592323010345838663995374553, −2.81199641290466401922104410182, −1.75257620037380458756357523666, 0.41606393121380592324780626731, 1.76960635658735112133165871690, 3.08565149026226327276546611746, 3.75915319077020106994138177379, 4.84080151645761567502847403087, 5.47141016339847157820555801417, 6.12022973901593065669869083454, 7.26622617955152909836348434444, 7.87159232572389719698473224918, 8.552469599948528403783907748349

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