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Properties

Label 1-385-385.18-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

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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: $\chi_{385} (18, \cdot )$
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 \( 1 + (0.951 - 0.309i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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