Properties

Label 1-20e2-400.67-r0-0-0
Degree $1$
Conductor $400$
Sign $-0.718 - 0.695i$
Analytic cond. $1.85759$
Root an. cond. $1.85759$
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)3-s + i·7-s + (−0.809 − 0.587i)9-s + (−0.587 − 0.809i)11-s + (−0.809 − 0.587i)13-s + (−0.951 + 0.309i)17-s + (−0.951 + 0.309i)19-s + (−0.951 − 0.309i)21-s + (0.587 + 0.809i)23-s + (0.809 − 0.587i)27-s + (−0.951 − 0.309i)29-s + (−0.309 − 0.951i)31-s + (0.951 − 0.309i)33-s + (−0.809 − 0.587i)37-s + (0.809 − 0.587i)39-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s + i·7-s + (−0.809 − 0.587i)9-s + (−0.587 − 0.809i)11-s + (−0.809 − 0.587i)13-s + (−0.951 + 0.309i)17-s + (−0.951 + 0.309i)19-s + (−0.951 − 0.309i)21-s + (0.587 + 0.809i)23-s + (0.809 − 0.587i)27-s + (−0.951 − 0.309i)29-s + (−0.309 − 0.951i)31-s + (0.951 − 0.309i)33-s + (−0.809 − 0.587i)37-s + (0.809 − 0.587i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(400\)    =    \(2^{4} \cdot 5^{2}\)
Sign: $-0.718 - 0.695i$
Analytic conductor: \(1.85759\)
Root analytic conductor: \(1.85759\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{400} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 400,\ (0:\ ),\ -0.718 - 0.695i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.03477129147 + 0.08584620168i\)
\(L(\frac12)\) \(\approx\) \(-0.03477129147 + 0.08584620168i\)
\(L(1)\) \(\approx\) \(0.6001577580 + 0.2230564469i\)
\(L(1)\) \(\approx\) \(0.6001577580 + 0.2230564469i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.309 + 0.951i)T \)
7 \( 1 + iT \)
11 \( 1 + (-0.587 - 0.809i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (-0.951 + 0.309i)T \)
19 \( 1 + (-0.951 + 0.309i)T \)
23 \( 1 + (0.587 + 0.809i)T \)
29 \( 1 + (-0.951 - 0.309i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.951 - 0.309i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (-0.587 + 0.809i)T \)
61 \( 1 + (-0.587 - 0.809i)T \)
67 \( 1 + (0.309 + 0.951i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.587 - 0.809i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.951 + 0.309i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.00519418542357168008602170509, −22.98306943788194170649586396905, −22.47425956017012973965940583837, −21.15001768679709548586177347311, −20.17222476943741571172822572903, −19.50074631418042096316532899980, −18.59580541194542873480313081244, −17.566904307955163314923027645251, −17.12430100837282190437213713367, −16.0966978005435175541581377598, −14.79455867403617276598754588753, −13.98545583969361005710898344401, −12.97396942691621260158206344623, −12.48500558059181531788193111355, −11.19371169159567185428181090459, −10.56245149522378759403345740369, −9.28278868963830098097353301790, −8.108433938878378920844492957470, −7.02024590049903183786336118740, −6.7676729955815723548091584960, −5.16104301796474388144369133737, −4.328272080495897732913200313, −2.653633563320866880391124474128, −1.68330874072960518499466455344, −0.052139305544489252621003972861, 2.26473388223929488838061723909, 3.27286377447495948260551649897, 4.50189558767203123596695086216, 5.524960332523654079991836204759, 6.12217519107845453865229133559, 7.74686927899083385493149349604, 8.80961225581643203869082884230, 9.49367143781906837380281105047, 10.67569474930090679559868391072, 11.273092854261694544721353710338, 12.37854756190246129949857838251, 13.28638967145255828882712379493, 14.71173535144982232983543836766, 15.24392113540061170602191323822, 16.00619179309130976266952324019, 17.01604300817404686922481089103, 17.76528520793772991655926974233, 18.8678398892687389965467690458, 19.71150235365179733764725179223, 20.8782071974674105363222696745, 21.50884148531391640881118291733, 22.18160047272948333566945469748, 22.97295352579495600165657218353, 24.09785656437651859335196741120, 24.87419261697944296692723872324

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