Properties

Label 1-407-407.159-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.555 - 0.831i$
Analytic cond. $1.89010$
Root an. cond. $1.89010$
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  + (0.104 + 0.994i)2-s + (0.669 − 0.743i)3-s + (−0.978 + 0.207i)4-s + (0.104 − 0.994i)5-s + (0.809 + 0.587i)6-s + (−0.978 + 0.207i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + 10-s + (−0.5 + 0.866i)12-s + (−0.913 − 0.406i)13-s + (−0.309 − 0.951i)14-s + (−0.669 − 0.743i)15-s + (0.913 − 0.406i)16-s + (−0.913 + 0.406i)17-s + (0.978 − 0.207i)18-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (0.669 − 0.743i)3-s + (−0.978 + 0.207i)4-s + (0.104 − 0.994i)5-s + (0.809 + 0.587i)6-s + (−0.978 + 0.207i)7-s + (−0.309 − 0.951i)8-s + (−0.104 − 0.994i)9-s + 10-s + (−0.5 + 0.866i)12-s + (−0.913 − 0.406i)13-s + (−0.309 − 0.951i)14-s + (−0.669 − 0.743i)15-s + (0.913 − 0.406i)16-s + (−0.913 + 0.406i)17-s + (0.978 − 0.207i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.555 - 0.831i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.555 - 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2546947559 - 0.4766521355i\)
\(L(\frac12)\) \(\approx\) \(0.2546947559 - 0.4766521355i\)
\(L(1)\) \(\approx\) \(0.8297249206 - 0.04279277143i\)
\(L(1)\) \(\approx\) \(0.8297249206 - 0.04279277143i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.104 + 0.994i)T \)
3 \( 1 + (0.669 - 0.743i)T \)
5 \( 1 + (0.104 - 0.994i)T \)
7 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (-0.913 - 0.406i)T \)
17 \( 1 + (-0.913 + 0.406i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 - T \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (-0.669 - 0.743i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.913 - 0.406i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−24.7257699667264937648385953898, −23.34165239009942669106683804596, −22.48554730494420035466332187306, −21.93437147290470398589029486191, −21.34551178821391593303019219574, −20.0796267007794025555802480697, −19.584574606666316020642353535254, −18.94502975823998075796451413540, −17.85552816616534768053479905288, −16.80864128733889418191183123371, −15.54896046765185138323200215786, −14.83192078633680499158159379870, −13.81030311144392590247009446428, −13.34859678419571529471470582493, −11.994324047026256852347671241154, −11.00764711024641211906116881656, −10.136391241341406817266921188201, −9.6485241437608148088907348148, −8.69339258700583067303022495947, −7.35225605126733681627263184837, −6.149176188199922737255170789182, −4.65201416563132545635750873931, −3.86819097124425650368930299746, −2.7656129200889787511004638603, −2.28544066209594195061517438950, 0.25806855242200355298143137695, 1.99597022239225459496403464844, 3.44891234437194413337004413365, 4.51949388293064008569389166243, 5.83071559192713656741123138352, 6.526373605595015924296301910, 7.6605730743805033723847141341, 8.4337189900517897201619707381, 9.23524185457570167417675715551, 10.01010731397296268649713425182, 12.18918819353059099078765148841, 12.7030371023220523297246906453, 13.33840268064716442992236339598, 14.30200232233180520349151152901, 15.25923353945330691829054083900, 16.05098572198039525248527738220, 17.03580575680698385002144065426, 17.716966911925716027028933288800, 18.81326500804945228288039195620, 19.57574535294334052830302698834, 20.364107940821099837300261877853, 21.60044832369093290452686245977, 22.48698701683470840057122732526, 23.48909589987212049284712609403, 24.28752980411804575781489064380

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