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Properties

Label 2-40e2-25.2-c0-0-0
Degree $2$
Conductor $1600$
Sign $0.982 + 0.187i$
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  + (−0.951 + 0.309i)5-s + (0.587 − 0.809i)9-s + (0.896 − 0.142i)13-s + (−0.142 + 0.278i)17-s + (0.809 − 0.587i)25-s + (1.11 + 0.363i)29-s + (−0.309 − 1.95i)37-s + (1.53 + 1.11i)41-s + (−0.309 + 0.951i)45-s − i·49-s + (0.809 + 1.58i)53-s + (1.53 − 1.11i)61-s + (−0.809 + 0.412i)65-s + (−0.278 + 1.76i)73-s + (−0.309 − 0.951i)81-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)5-s + (0.587 − 0.809i)9-s + (0.896 − 0.142i)13-s + (−0.142 + 0.278i)17-s + (0.809 − 0.587i)25-s + (1.11 + 0.363i)29-s + (−0.309 − 1.95i)37-s + (1.53 + 1.11i)41-s + (−0.309 + 0.951i)45-s − i·49-s + (0.809 + 1.58i)53-s + (1.53 − 1.11i)61-s + (−0.809 + 0.412i)65-s + (−0.278 + 1.76i)73-s + (−0.309 − 0.951i)81-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $0.982 + 0.187i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ 0.982 + 0.187i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.025586588\)
\(L(\frac12)\) \(\approx\) \(1.025586588\)
\(L(1)\) 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 \( 1 + (0.951 - 0.309i)T \)
good3 \( 1 + (-0.587 + 0.809i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.896 + 0.142i)T + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (0.142 - 0.278i)T + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.951 - 0.309i)T^{2} \)
29 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.309 + 1.95i)T + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (-1.53 - 1.11i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.587 - 0.809i)T^{2} \)
53 \( 1 + (-0.809 - 1.58i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.278 - 1.76i)T + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T^{2} \)
89 \( 1 + (0.690 + 0.951i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2} \)
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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

−9.520568005641488839881160474243, −8.708830992038438053970039278466, −8.024869064796451083956294233236, −7.11297956890935470564580721576, −6.52480319076295896577940422337, −5.55469408517207116764949506139, −4.24002720633802896141332006660, −3.80850965172574500538528752456, −2.71906808992205826794541362401, −1.05758427790037695615439592140, 1.22417954720436596909678132503, 2.66816079362897159731108229960, 3.85858501934847749625421286484, 4.52798347697426912472995227353, 5.38810827549034111809626390686, 6.55645950040759757831858631367, 7.28804353718534806734788595838, 8.152371229130739577164399849995, 8.593011549956118793529923554319, 9.620963737288975775170874060678

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