↑

Properties

Label 1-385-385.107-r0-0-0
Degree $1$
Conductor $385$
Sign $-0.893 + 0.448i$
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  + (0.207 + 0.978i)2-s + (−0.994 + 0.104i)3-s + (−0.913 + 0.406i)4-s + (−0.309 − 0.951i)6-s + (−0.587 − 0.809i)8-s + (0.978 − 0.207i)9-s + (0.866 − 0.5i)12-s + (0.951 + 0.309i)13-s + (0.669 − 0.743i)16-s + (−0.207 + 0.978i)17-s + (0.406 + 0.913i)18-s + (0.913 + 0.406i)19-s + (−0.866 + 0.5i)23-s + (0.669 + 0.743i)24-s + (−0.104 + 0.994i)26-s + (−0.951 + 0.309i)27-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.994 + 0.104i)3-s + (−0.913 + 0.406i)4-s + (−0.309 − 0.951i)6-s + (−0.587 − 0.809i)8-s + (0.978 − 0.207i)9-s + (0.866 − 0.5i)12-s + (0.951 + 0.309i)13-s + (0.669 − 0.743i)16-s + (−0.207 + 0.978i)17-s + (0.406 + 0.913i)18-s + (0.913 + 0.406i)19-s + (−0.866 + 0.5i)23-s + (0.669 + 0.743i)24-s + (−0.104 + 0.994i)26-s + (−0.951 + 0.309i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(385\)    =    \(5 \cdot 7 \cdot 11\)
Sign: $-0.893 + 0.448i$
Analytic conductor: \(1.78793\)
Root analytic conductor: \(1.78793\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{385} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 385,\ (0:\ ),\ -0.893 + 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1823554333 + 0.7693956675i\)
\(L(\frac12)\) \(\approx\) \(0.1823554333 + 0.7693956675i\)
\(L(1)\) \(\approx\) \(0.5985948586 + 0.5033065219i\)
\(L(1)\) \(\approx\) \(0.5985948586 + 0.5033065219i\)

Euler product

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

−23.941143277279208081823117503603, −22.86973650300998911324635302757, −22.533601137227136731073948329007, −21.60477437234363548060850282026, −20.67248090580849713598651065203, −19.95527692495451919033208706347, −18.580309335020424950167278474700, −18.294169367147755443844953027036, −17.35158016079288524536508826039, −16.23415423486130052660832960253, −15.344884206416347948207268703331, −13.91083470591478652754672214212, −13.28102530637111794800505978236, −12.2313168898822056174630883768, −11.54417835982243608974196798192, −10.770217229390224844373812884997, −9.90166228358692665947394725206, −8.88495778445665617612955012477, −7.52858324338705409211097827679, −6.206765812870923806544111555420, −5.32572428540363638470650117600, −4.39036112031989860709162206711, −3.23446632889365233683509095632, −1.81057808555803264615138771009, −0.569984024128416959381434132807, 1.35132151924990122204261471358, 3.59363327977909657788807468426, 4.41388962097993412290659727096, 5.639618250970807416270707553668, 6.15505405993375616700855840019, 7.21052344407368292904287797047, 8.21402420001415943281167525061, 9.373808319766040366828093342927, 10.355006094412310159406112214603, 11.504941160346696555049036129472, 12.42686085405673984099254469065, 13.34436106231098996409354678095, 14.2524134724805367055851383110, 15.48846308854473777600809957003, 16.00205130996128103546826735068, 16.86274890246056087298431953611, 17.716361877838786111952211450095, 18.32536333421518753669574593149, 19.35037065249332280641111906029, 20.961767681832503393553346067349, 21.59937321234262713148901444259, 22.64405378589797375302386894190, 23.05031522205660795170906008019, 24.1272564449096020823871760065, 24.48784684686668406653682645468

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