Properties

Label 1-385-385.299-r0-0-0
Degree $1$
Conductor $385$
Sign $-0.968 + 0.250i$
Analytic cond. $1.78793$
Root an. cond. $1.78793$
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  + (−0.104 + 0.994i)2-s + (0.669 + 0.743i)3-s + (−0.978 − 0.207i)4-s + (−0.809 + 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.5 − 0.866i)12-s + (0.809 + 0.587i)13-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (−0.978 − 0.207i)18-s + (−0.978 + 0.207i)19-s + (0.5 + 0.866i)23-s + (0.913 − 0.406i)24-s + (−0.669 + 0.743i)26-s + (−0.809 + 0.587i)27-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)2-s + (0.669 + 0.743i)3-s + (−0.978 − 0.207i)4-s + (−0.809 + 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.5 − 0.866i)12-s + (0.809 + 0.587i)13-s + (0.913 + 0.406i)16-s + (0.104 + 0.994i)17-s + (−0.978 − 0.207i)18-s + (−0.978 + 0.207i)19-s + (0.5 + 0.866i)23-s + (0.913 − 0.406i)24-s + (−0.669 + 0.743i)26-s + (−0.809 + 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.968 + 0.250i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ -0.968 + 0.250i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1601453368 + 1.256921892i\)
\(L(\frac12)\) \(\approx\) \(0.1601453368 + 1.256921892i\)
\(L(1)\) \(\approx\) \(0.7144430014 + 0.8351736127i\)
\(L(1)\) \(\approx\) \(0.7144430014 + 0.8351736127i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.104 + 0.994i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.309 - 0.951i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (-0.669 + 0.743i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 + (-0.913 + 0.406i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (0.978 + 0.207i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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

−23.98080958213963736336777161441, −23.155393122126948108661882296552, −22.37846220932159453410161580026, −21.10011411539652087017963374150, −20.57608546492559805623961653419, −19.77305286735337808596752297391, −18.90575295309509008736979585714, −18.244854141767213796380837408872, −17.52160702038860035626453407422, −16.23902154376672528995959686721, −14.82056206994209029334162407761, −14.14120251772494978763584221400, −13.00601133461605157402950651630, −12.734466223964239761315936218155, −11.48089588329983834477041758627, −10.66792090384611454963433485683, −9.41696811714667524759295734908, −8.6941518569879479130142388541, −7.82994204212449233934332883856, −6.644623218537368734458020803131, −5.267473115028767979774702175348, −3.904974803646053173344887332028, −2.97132286597196723145331906940, −2.00212980528734245652547982873, −0.754601512029795221302766888963, 1.766601921619578154458598370332, 3.549315594737827044250127062021, 4.22634630450857535774005596270, 5.39858501213933145860266001350, 6.39970911664871039944593829516, 7.61826008771001256778636655224, 8.51915502497098902717234330462, 9.16666587661079954457801893403, 10.18260390708237656672743031398, 11.0995733290302614215055669177, 12.77934975762843723383290950498, 13.61409285581105110818051202475, 14.47230389664730058939573317974, 15.2159618397615128169143029307, 15.95466840837101074747164049376, 16.81305557152205021418542076149, 17.59592914060798956344650540278, 19.0017681221431285399331934387, 19.307301636865079300483850621593, 20.76645105123578251986833827810, 21.45708827511829419576359144256, 22.33306678196048849368651777051, 23.33197810093113595253830697670, 24.0801164539249747374031269537, 25.16718749541276323688631540696

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