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Properties

Label 2-40e2-100.39-c0-0-0
Degree $2$
Conductor $1600$
Sign $0.187 - 0.982i$
Analytic cond. $0.798504$
Root an. cond. $0.893590$
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 + (−1.11 + 1.53i)13-s + (−1.11 + 0.363i)17-s + (−0.809 + 0.587i)25-s + (0.5 − 1.53i)29-s + (0.690 − 0.951i)37-s + (0.5 + 0.363i)41-s + (−0.309 + 0.951i)45-s − 49-s + (1.80 + 0.587i)53-s + (−0.5 + 0.363i)61-s + (−1.80 − 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 + (−1.11 + 1.53i)13-s + (−1.11 + 0.363i)17-s + (−0.809 + 0.587i)25-s + (0.5 − 1.53i)29-s + (0.690 − 0.951i)37-s + (0.5 + 0.363i)41-s + (−0.309 + 0.951i)45-s − 49-s + (1.80 + 0.587i)53-s + (−0.5 + 0.363i)61-s + (−1.80 − 0.587i)65-s + (1.11 + 1.53i)73-s + (0.309 + 0.951i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.187 - 0.982i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (639, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.187 - 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.102121899\)
\(L(\frac12)\) \(\approx\) \(1.102121899\)
\(L(1)\) 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 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
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 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
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 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
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

−9.767131372936824016960834342938, −9.219991429740028145507424713316, −8.047691475726704162465960262546, −7.19979416116004875302746828184, −6.73618487811764035526541548003, −5.85777720593628579012950864116, −4.57626925998474628893930535386, −4.08743388756066306345171605385, −2.52338869628825071127256200998, −1.97298034756322707639412586458, 0.878165958749698992194830907739, 2.26072679012477931302542045470, 3.44224887904618454575163181064, 4.67613391457138330348827178304, 5.08505893862091891092250262667, 6.17661108742695825407368106777, 7.05838159153716982083698927831, 7.87160233343721840830522931715, 8.718042497599433121095432084682, 9.444724700092636256444720249618

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