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Properties

Label 2-800-25.22-c0-0-0
Degree $2$
Conductor $800$
Sign $0.750 - 0.661i$
Analytic cond. $0.399252$
Root an. cond. $0.631863$
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 + (1.76 + 0.896i)13-s + (−0.896 − 0.142i)17-s + (−0.309 + 0.951i)25-s + (1.11 − 1.53i)29-s + (−0.809 + 1.58i)37-s + (−0.363 − 1.11i)41-s + (−0.809 − 0.587i)45-s − i·49-s + (0.309 − 0.0489i)53-s + (0.363 − 1.11i)61-s + (0.309 + 1.95i)65-s + (0.142 + 0.278i)73-s + (0.809 − 0.587i)81-s + ⋯
L(s)  = 1  + (0.587 + 0.809i)5-s + (−0.951 + 0.309i)9-s + (1.76 + 0.896i)13-s + (−0.896 − 0.142i)17-s + (−0.309 + 0.951i)25-s + (1.11 − 1.53i)29-s + (−0.809 + 1.58i)37-s + (−0.363 − 1.11i)41-s + (−0.809 − 0.587i)45-s − i·49-s + (0.309 − 0.0489i)53-s + (0.363 − 1.11i)61-s + (0.309 + 1.95i)65-s + (0.142 + 0.278i)73-s + (0.809 − 0.587i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.750 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(800\)    =    \(2^{5} \cdot 5^{2}\)
Sign: $0.750 - 0.661i$
Analytic conductor: \(0.399252\)
Root analytic conductor: \(0.631863\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{800} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 800,\ (\ :0),\ 0.750 - 0.661i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.038146927\)
\(L(\frac12)\) \(\approx\) \(1.038146927\)
\(L(1)\) 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 \( 1 + (-1.76 - 0.896i)T + (0.587 + 0.809i)T^{2} \)
17 \( 1 + (0.896 + 0.142i)T + (0.951 + 0.309i)T^{2} \)
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 \( 1 + (0.809 - 1.58i)T + (-0.587 - 0.809i)T^{2} \)
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 \( 1 + (-0.309 + 0.0489i)T + (0.951 - 0.309i)T^{2} \)
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 \( 1 + (-0.142 - 0.278i)T + (-0.587 + 0.809i)T^{2} \)
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 \( 1 + (0.278 + 1.76i)T + (-0.951 + 0.309i)T^{2} \)
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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

−10.66707562535122969719448446436, −9.788045261971032128113496986233, −8.760563975492427681426651636507, −8.251579583777890979605814305287, −6.80551206437546244944942253702, −6.34689387102231706495389691178, −5.42658034022304874998842089651, −4.10907192098185428545839539245, −3.00046108852257782490590578472, −1.89918481493679623190188232161, 1.25083029184555561704519537629, 2.80848756553467762907928258250, 3.95181827442789150578269292054, 5.20756422265877943509939286550, 5.90173597852741842246532603760, 6.68904461783371233233676843717, 8.192198531674797190113666619432, 8.674928797498916546881865912139, 9.291000955799459734747056853104, 10.54641437560242038325958480526

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