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Properties

Label 1-840-840.419-r1-0-0
Degree $1$
Conductor $840$
Sign $1$
Analytic cond. $90.2705$
Root an. cond. $90.2705$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 11-s − 13-s − 17-s − 19-s − 23-s + 29-s + 31-s + 37-s + 41-s − 43-s + 47-s − 53-s + 59-s + 61-s − 67-s + 71-s + 73-s − 79-s − 83-s + 89-s + 97-s + ⋯
L(s)  = 1  − 11-s − 13-s − 17-s − 19-s − 23-s + 29-s + 31-s + 37-s + 41-s − 43-s + 47-s − 53-s + 59-s + 61-s − 67-s + 71-s + 73-s − 79-s − 83-s + 89-s + 97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(840\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(90.2705\)
Root analytic conductor: \(90.2705\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{840} (419, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 840,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.226980037\)
\(L(\frac12)\) \(\approx\) \(1.226980037\)
\(L(1)\) \(\approx\) \(0.8671619566\)
\(L(1)\) \(\approx\) \(0.8671619566\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 + T \)
37 \( 1 + T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 - T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + T \)
97 \( 1 + T \)
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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

−21.8275200004266104129113874406, −21.27119025589043022260930878939, −20.25755891306168317954308813613, −19.59814060728535599742470570599, −18.7757440248558494393385467482, −17.83090204741790949270081143423, −17.2914596715650306131957527491, −16.21230942325833836525055326591, −15.52828758601783531607749489143, −14.729412460163290148035448229067, −13.80471728489882390662542925899, −12.966750476510620314132802216387, −12.24852208848277077037804294045, −11.26454121255882272589454275158, −10.372108296152060261887761209826, −9.714000378867945789114010253654, −8.54396411916825415099145308969, −7.87850741909143007556764279109, −6.84924458091286980919694401898, −5.986370789477445191574867681472, −4.86511848161247726953639003173, −4.20095298768179179429389977064, −2.74148938984140444977286539863, −2.13535048585682735929798903533, −0.50605574805019718502311711236, 0.50605574805019718502311711236, 2.13535048585682735929798903533, 2.74148938984140444977286539863, 4.20095298768179179429389977064, 4.86511848161247726953639003173, 5.986370789477445191574867681472, 6.84924458091286980919694401898, 7.87850741909143007556764279109, 8.54396411916825415099145308969, 9.714000378867945789114010253654, 10.372108296152060261887761209826, 11.26454121255882272589454275158, 12.24852208848277077037804294045, 12.966750476510620314132802216387, 13.80471728489882390662542925899, 14.729412460163290148035448229067, 15.52828758601783531607749489143, 16.21230942325833836525055326591, 17.2914596715650306131957527491, 17.83090204741790949270081143423, 18.7757440248558494393385467482, 19.59814060728535599742470570599, 20.25755891306168317954308813613, 21.27119025589043022260930878939, 21.8275200004266104129113874406

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