↑

Properties

Label 1-1380-1380.479-r1-0-0
Degree $1$
Conductor $1380$
Sign $0.0132 - 0.999i$
Analytic cond. $148.301$
Root an. cond. $148.301$
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  + (0.959 − 0.281i)7-s + (0.654 + 0.755i)11-s + (0.959 + 0.281i)13-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (0.142 − 0.989i)29-s + (−0.841 − 0.540i)31-s + (0.415 − 0.909i)37-s + (−0.415 − 0.909i)41-s + (−0.841 + 0.540i)43-s − 47-s + (0.841 − 0.540i)49-s + (0.959 − 0.281i)53-s + (−0.959 − 0.281i)59-s + (−0.841 − 0.540i)61-s + ⋯
L(s)  = 1  + (0.959 − 0.281i)7-s + (0.654 + 0.755i)11-s + (0.959 + 0.281i)13-s + (0.142 − 0.989i)17-s + (−0.142 − 0.989i)19-s + (0.142 − 0.989i)29-s + (−0.841 − 0.540i)31-s + (0.415 − 0.909i)37-s + (−0.415 − 0.909i)41-s + (−0.841 + 0.540i)43-s − 47-s + (0.841 − 0.540i)49-s + (0.959 − 0.281i)53-s + (−0.959 − 0.281i)59-s + (−0.841 − 0.540i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0132 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0132 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1380\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.0132 - 0.999i$
Analytic conductor: \(148.301\)
Root analytic conductor: \(148.301\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1380} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1380,\ (1:\ ),\ 0.0132 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.657608136 - 1.635735170i\)
\(L(\frac12)\) \(\approx\) \(1.657608136 - 1.635735170i\)
\(L(1)\) \(\approx\) \(1.238674893 - 0.2167507069i\)
\(L(1)\) \(\approx\) \(1.238674893 - 0.2167507069i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good7 \( 1 + (0.959 - 0.281i)T \)
11 \( 1 + (0.654 + 0.755i)T \)
13 \( 1 + (0.959 + 0.281i)T \)
17 \( 1 + (0.142 - 0.989i)T \)
19 \( 1 + (-0.142 - 0.989i)T \)
29 \( 1 + (0.142 - 0.989i)T \)
31 \( 1 + (-0.841 - 0.540i)T \)
37 \( 1 + (0.415 - 0.909i)T \)
41 \( 1 + (-0.415 - 0.909i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 - T \)
53 \( 1 + (0.959 - 0.281i)T \)
59 \( 1 + (-0.959 - 0.281i)T \)
61 \( 1 + (-0.841 - 0.540i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (-0.654 + 0.755i)T \)
73 \( 1 + (0.142 + 0.989i)T \)
79 \( 1 + (-0.959 - 0.281i)T \)
83 \( 1 + (0.415 - 0.909i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (0.415 + 0.909i)T \)
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

−20.89366277796296335687423908677, −20.11448087661762062147741712965, −19.32816636023384883087364365538, −18.34456425704211006749005040168, −18.08130283257360289011490776821, −16.83245654987577548863262576952, −16.54554230772026723691952901671, −15.3607594692781050034094234538, −14.72924783271368162449253632180, −14.06280456246926575628803159618, −13.21698543988398960130633220725, −12.306788017928304461837469309255, −11.53446948262260814404194652371, −10.84302265108553041481725582021, −10.13083572049790012751498815291, −8.7972535975312236410206928905, −8.48917455163600124049322839841, −7.642304252866189533320983052243, −6.41908082529672056062508924196, −5.828690173502575596123397576553, −4.90125109263728641091637510411, −3.84518205602021911681794389154, −3.17231839118293510447957017028, −1.67770214190538152849189675231, −1.22791388670739298486255686203, 0.448541395377988602856064225866, 1.50944722240242500433393508143, 2.33705287720192013475850300176, 3.636970456679083178325900312189, 4.44113045940807476589879328757, 5.154837534966349571934578213256, 6.26537521974797675500810601947, 7.11762986387058025227520450009, 7.81814533370650308568598971852, 8.823270615603089635764328740280, 9.44320068704983114690788380219, 10.45131276941510171447280042229, 11.42789846920342350686330033670, 11.659465705786444422226897980, 12.87148695053409542445719322334, 13.65190922599820035533271767041, 14.34082353154923356889723513668, 15.08366429890058009403440618751, 15.86047176907327757677010169239, 16.77145679581085179477810006251, 17.4938589615796705989645817321, 18.12785673633978948924579685414, 18.82523507668459718746200091660, 19.97156892491863591374386487737, 20.33640491222239402078071451435

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