Properties

Label 1-1400-1400.461-r1-0-0
Degree $1$
Conductor $1400$
Sign $0.728 + 0.684i$
Analytic cond. $150.450$
Root an. cond. $150.450$
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  + (−0.809 − 0.587i)3-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)17-s + (−0.809 + 0.587i)19-s + (0.309 − 0.951i)23-s + (0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)33-s + (−0.309 − 0.951i)37-s + (0.309 − 0.951i)39-s + (−0.309 − 0.951i)41-s − 43-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)3-s + (0.309 + 0.951i)9-s + (−0.309 + 0.951i)11-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)17-s + (−0.809 + 0.587i)19-s + (0.309 − 0.951i)23-s + (0.309 − 0.951i)27-s + (0.809 + 0.587i)29-s + (0.809 − 0.587i)31-s + (0.809 − 0.587i)33-s + (−0.309 − 0.951i)37-s + (0.309 − 0.951i)39-s + (−0.309 − 0.951i)41-s − 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.728 + 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1400\)    =    \(2^{3} \cdot 5^{2} \cdot 7\)
Sign: $0.728 + 0.684i$
Analytic conductor: \(150.450\)
Root analytic conductor: \(150.450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1400} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1400,\ (1:\ ),\ 0.728 + 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.275086520 + 0.5048424672i\)
\(L(\frac12)\) \(\approx\) \(1.275086520 + 0.5048424672i\)
\(L(1)\) \(\approx\) \(0.8482965079 + 0.01580971261i\)
\(L(1)\) \(\approx\) \(0.8482965079 + 0.01580971261i\)

Euler product

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

−20.755782956525558038838528539655, −19.742621611232344515571663116338, −18.96721242852042322156063967582, −18.147945503169270934429771935, −17.317934572936901620304650065102, −16.877663375302043221904553580847, −15.873322003907805163276487664, −15.43002560050535432607074400270, −14.62350867912804227788964091495, −13.4729679801199061266827525690, −12.8979714159611228092518316682, −11.834147812421055053577059338244, −11.31814877025608362744679427530, −10.261316445294965394531648836762, −10.1105198899457127048620981736, −8.71416972595668829828433314865, −8.19594878882604043863485604934, −6.95103524558104490630237950637, −6.080758337570389175257212533661, −5.471880683167241979603082353338, −4.64846511045669928048008699256, −3.5739792989677011073671710985, −2.92076459921437032651425507248, −1.28280340115032555070479579241, −0.43106323093623969530565032050, 0.7528930433796518360264976760, 1.78462462577865785598603145115, 2.588742457121206843780816295576, 4.06851293941470991113972894281, 4.79649220822213687781868667644, 5.67409142354327936133672268827, 6.60023562622019449379902303252, 7.15224203131632059200748778260, 8.0487768360960003018120425767, 8.97931250018835605528936862166, 10.11583042023605166322563606304, 10.60608124938922751416294877589, 11.65488117537749413360312337711, 12.24970726660557811967937561186, 12.82740532573140647118868922749, 13.82137058304016804836512316084, 14.47203463964820054465850574792, 15.53147366151678283784589913514, 16.3417139236514072199592168215, 16.95356988977657363956643021270, 17.65677080324527367058159981016, 18.62737079543812928102240924179, 18.81148327174384433299007074891, 19.87142875607720735659419484278, 20.83739128196483789468285290160

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