↑

Properties

Label 1-385-385.257-r0-0-0
Degree $1$
Conductor $385$
Sign $-0.960 + 0.278i$
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  + (0.743 + 0.669i)2-s + (0.406 + 0.913i)3-s + (0.104 + 0.994i)4-s + (−0.309 + 0.951i)6-s + (−0.587 + 0.809i)8-s + (−0.669 + 0.743i)9-s + (−0.866 + 0.5i)12-s + (−0.951 + 0.309i)13-s + (−0.978 + 0.207i)16-s + (0.743 − 0.669i)17-s + (−0.994 + 0.104i)18-s + (−0.104 + 0.994i)19-s + (−0.866 + 0.5i)23-s + (−0.978 − 0.207i)24-s + (−0.913 − 0.406i)26-s + (−0.951 − 0.309i)27-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.406 + 0.913i)3-s + (0.104 + 0.994i)4-s + (−0.309 + 0.951i)6-s + (−0.587 + 0.809i)8-s + (−0.669 + 0.743i)9-s + (−0.866 + 0.5i)12-s + (−0.951 + 0.309i)13-s + (−0.978 + 0.207i)16-s + (0.743 − 0.669i)17-s + (−0.994 + 0.104i)18-s + (−0.104 + 0.994i)19-s + (−0.866 + 0.5i)23-s + (−0.978 − 0.207i)24-s + (−0.913 − 0.406i)26-s + (−0.951 − 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.960 + 0.278i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ -0.960 + 0.278i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2691394898 + 1.893128153i\)
\(L(\frac12)\) \(\approx\) \(0.2691394898 + 1.893128153i\)
\(L(1)\) \(\approx\) \(1.020003574 + 1.200849870i\)
\(L(1)\) \(\approx\) \(1.020003574 + 1.200849870i\)

Euler product

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

−24.09881971340967870386580884433, −23.27471914658581536817418015000, −22.40310832542389896064764510365, −21.474135475445767683954408648966, −20.57074684151199454507797564773, −19.5743160598727006429783174849, −19.32110755383612253112278084743, −18.13984801681279513576763555674, −17.36304617540967846093051736543, −15.88734818730723096747387836911, −14.76976639251962003063456698899, −14.24268764308244530018024023776, −13.25836342381055385813255576183, −12.450564379269887176173557961223, −11.89267713661304087754255491241, −10.6731666006376611467796865027, −9.70007658198413090681217712092, −8.56097947349876105345354560062, −7.38857207240437395046954199465, −6.407860477808077781045543669739, −5.42178359047656380834666787021, −4.179042059784001666429365693146, −2.93855611509054599683838017154, −2.18098179657878859403679430291, −0.84241178163368830907854580389, 2.347579262879084551412221570149, 3.36062657140092397178486897120, 4.36089623940426665664718519101, 5.17688767755303090058398912136, 6.19221815733112347360777078475, 7.54129614809235406556176211736, 8.248348182246546551418429751864, 9.44218956946023695426546374652, 10.27392274808593239136242784035, 11.684636518168036315598843092046, 12.33237673906097015708841852931, 13.85333286420997963760559793016, 14.15759879312213742626184128233, 15.18871156218704448573753104638, 15.89869896272800277269388454124, 16.73437955746065727552775436141, 17.4374291473931010779539206571, 18.840188095398588785681794402046, 19.96293171937258006134211970674, 20.851386250854545371325668543192, 21.48954108952212531660095911904, 22.32911437551943917933661374087, 23.02108673735666677534816303109, 24.03356982343939078961647117284, 25.09047205363536415777102561247

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