↑

Properties

Label 1-2520-2520.1973-r1-0-0
Degree $1$
Conductor $2520$
Sign $0.949 + 0.313i$
Analytic cond. $270.811$
Root an. cond. $270.811$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)13-s − i·17-s − 19-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s − i·37-s + (−0.5 + 0.866i)41-s + (−0.866 + 0.5i)43-s + (−0.866 + 0.5i)47-s − i·53-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)61-s + (−0.866 − 0.5i)67-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)11-s + (−0.866 − 0.5i)13-s − i·17-s − 19-s + (0.866 + 0.5i)23-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s − i·37-s + (−0.5 + 0.866i)41-s + (−0.866 + 0.5i)43-s + (−0.866 + 0.5i)47-s − i·53-s + (−0.5 + 0.866i)59-s + (−0.5 − 0.866i)61-s + (−0.866 − 0.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2520 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2520\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.949 + 0.313i$
Analytic conductor: \(270.811\)
Root analytic conductor: \(270.811\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2520} (1973, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2520,\ (1:\ ),\ 0.949 + 0.313i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.237635397 + 0.1991513553i\)
\(L(\frac12)\) \(\approx\) \(1.237635397 + 0.1991513553i\)
\(L(1)\) \(\approx\) \(0.8900170081 - 0.06135771474i\)
\(L(1)\) \(\approx\) \(0.8900170081 - 0.06135771474i\)

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 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 - iT \)
19 \( 1 - T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - iT \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 + (-0.866 + 0.5i)T \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 - iT \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 - T \)
73 \( 1 - iT \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.866 + 0.5i)T \)
89 \( 1 - T \)
97 \( 1 + (0.866 - 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.34807609857657746223592241107, −18.587185779888321781989836185737, −17.618252740120050662123163282743, −17.2216619499667685587662158438, −16.47761983782241994373889704320, −15.52802269174803084213483212795, −14.93985888334863950247377994047, −14.40123776025045732210794926665, −13.387914270635767069321564793764, −12.68658133545510005863582146386, −12.18329531227107125431624136102, −11.28269645484014339489067486632, −10.28956010448854710397947750749, −10.05287656034609874288597015990, −8.88083649093009118162449949562, −8.35634821834252927306662265694, −7.33005416599871781540910410913, −6.78680541622839239184246940373, −5.905797863412553624584097829655, −4.85113486962745776387460120388, −4.42138752872923838304453778073, −3.34807296227260339484945767131, −2.301075867638482915116983920528, −1.75131056907823179868041872343, −0.341146106782012182360044265684, 0.52720228585713626987689151661, 1.57596102129342307137616274119, 2.913986593388190513272913687235, 3.04676989410381683242417601745, 4.542915435151040133788479830336, 5.01184831085659428592133861667, 5.96245465919127306768931025724, 6.73620138995909088526326826456, 7.62112581694668440912333026370, 8.25945442045354338829196490463, 9.09506478955269357303689133492, 9.88274504046589048387821249613, 10.61659008712985518568766705339, 11.35337398631543734621289920697, 12.05162929156105705057049846621, 12.976647156910950346040459849269, 13.46190993711692061176729797770, 14.30600018263728462913544946943, 15.11868973268971620861973585798, 15.61498437716260192878633939125, 16.693075315097598906268730591874, 16.93320396914442715088268515319, 18.01769710234873277348558565742, 18.52323991208454730090657470037, 19.358573063945502054558831428115

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