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Properties

Label 2-40e2-25.3-c0-0-0
Degree $2$
Conductor $1600$
Sign $0.968 - 0.248i$
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  + (0.587 + 0.809i)5-s + (0.951 − 0.309i)9-s + (0.142 − 0.278i)13-s + (0.278 − 1.76i)17-s + (−0.309 + 0.951i)25-s + (−1.11 + 1.53i)29-s + (0.809 + 0.412i)37-s + (0.363 + 1.11i)41-s + (0.809 + 0.587i)45-s + i·49-s + (−0.309 − 1.95i)53-s + (0.363 − 1.11i)61-s + (0.309 − 0.0489i)65-s + (−1.76 + 0.896i)73-s + (0.809 − 0.587i)81-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)5-s + (0.951 − 0.309i)9-s + (0.142 − 0.278i)13-s + (0.278 − 1.76i)17-s + (−0.309 + 0.951i)25-s + (−1.11 + 1.53i)29-s + (0.809 + 0.412i)37-s + (0.363 + 1.11i)41-s + (0.809 + 0.587i)45-s + i·49-s + (−0.309 − 1.95i)53-s + (0.363 − 1.11i)61-s + (0.309 − 0.0489i)65-s + (−1.76 + 0.896i)73-s + (0.809 − 0.587i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.327881752\)
\(L(\frac12)\) \(\approx\) \(1.327881752\)
\(L(1)\) 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 \( 1 + (-0.587 - 0.809i)T \)
good3 \( 1 + (-0.951 + 0.309i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.142 + 0.278i)T + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (-0.278 + 1.76i)T + (-0.951 - 0.309i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.587 - 0.809i)T^{2} \)
29 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 - 0.412i)T + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.309 + 1.95i)T + (-0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.951 - 0.309i)T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.951 + 0.309i)T^{2} \)
89 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.896 + 0.142i)T + (0.951 - 0.309i)T^{2} \)
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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

−9.658672906600701360256114678485, −9.123126041769436091195164882741, −7.83663249872692499164324901402, −7.14366903057640265618173494550, −6.57059587271854711079377851298, −5.55971064093244714301434776206, −4.73094064850979258543967327394, −3.51703807664919878822579835544, −2.71391936257685135070765633816, −1.42135185573543735139228365539, 1.38597062127487671088004263455, 2.24334037260351118150820148139, 3.93417104555978478948080325107, 4.40987238835565119964897941537, 5.64216502307647510022793888532, 6.10386728024586203504632720484, 7.27382616876888295029953800214, 8.023529659432208136152386556458, 8.821656661151736415302074188965, 9.588208979648006866376231482943

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