↑

Properties

Label 1-407-407.367-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.261 + 0.965i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.882 + 0.469i)2-s + (−0.719 + 0.694i)3-s + (0.559 − 0.829i)4-s + (0.848 − 0.529i)5-s + (0.309 − 0.951i)6-s + (−0.997 − 0.0697i)7-s + (−0.104 + 0.994i)8-s + (0.0348 − 0.999i)9-s + (−0.5 + 0.866i)10-s + (0.173 + 0.984i)12-s + (−0.615 − 0.788i)13-s + (0.913 − 0.406i)14-s + (−0.241 + 0.970i)15-s + (−0.374 − 0.927i)16-s + (−0.615 + 0.788i)17-s + (0.438 + 0.898i)18-s + ⋯
L(s)  = 1  + (−0.882 + 0.469i)2-s + (−0.719 + 0.694i)3-s + (0.559 − 0.829i)4-s + (0.848 − 0.529i)5-s + (0.309 − 0.951i)6-s + (−0.997 − 0.0697i)7-s + (−0.104 + 0.994i)8-s + (0.0348 − 0.999i)9-s + (−0.5 + 0.866i)10-s + (0.173 + 0.984i)12-s + (−0.615 − 0.788i)13-s + (0.913 − 0.406i)14-s + (−0.241 + 0.970i)15-s + (−0.374 − 0.927i)16-s + (−0.615 + 0.788i)17-s + (0.438 + 0.898i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.261 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.261 + 0.965i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.261 + 0.965i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3076608258 + 0.4020051074i\)
\(L(\frac12)\) \(\approx\) \(0.3076608258 + 0.4020051074i\)
\(L(1)\) \(\approx\) \(0.5078763557 + 0.2026387997i\)
\(L(1)\) \(\approx\) \(0.5078763557 + 0.2026387997i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.882 + 0.469i)T \)
3 \( 1 + (-0.719 + 0.694i)T \)
5 \( 1 + (0.848 - 0.529i)T \)
7 \( 1 + (-0.997 - 0.0697i)T \)
13 \( 1 + (-0.615 - 0.788i)T \)
17 \( 1 + (-0.615 + 0.788i)T \)
19 \( 1 + (-0.719 + 0.694i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.913 + 0.406i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (0.961 + 0.275i)T \)
43 \( 1 + T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (0.848 + 0.529i)T \)
59 \( 1 + (-0.241 + 0.970i)T \)
61 \( 1 + (0.0348 + 0.999i)T \)
67 \( 1 + (0.766 + 0.642i)T \)
71 \( 1 + (-0.882 - 0.469i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.374 + 0.927i)T \)
83 \( 1 + (-0.615 + 0.788i)T \)
89 \( 1 + (0.766 - 0.642i)T \)
97 \( 1 + (-0.978 - 0.207i)T \)
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

−24.3783847960229342815430312955, −22.994261105179864193243595910080, −22.13837650900612823107576677281, −21.70597879003432585841149796971, −20.45099662298022680056283864705, −19.2556144163091995470280249828, −18.94941342887333062528059532762, −17.94710229918389743721713648392, −17.3198586600509128401672812966, −16.53987866924295727859613584313, −15.65433556973089038956007405678, −14.05294964788364087665451662522, −13.15124480434685195919245167827, −12.38756533106978478872126493507, −11.43270626347657769828427666536, −10.560319773348462702536644477245, −9.73341894888939541723079738896, −8.88838170324301305785529638286, −7.43057591356082360475860611571, −6.66015242765448791631215154094, −6.10980867725565796064609557296, −4.43576942809558668181216553132, −2.621196488891077103538530491316, −2.19333932530561111744982781387, −0.49314775597623890990225397063, 1.08960721205729337886888152208, 2.63864155943118668036922401215, 4.27360299726996159829872251391, 5.58978875886898176703947226210, 6.02804455363073853605007067198, 7.04623829509212938206911567159, 8.531388713238412767682496661925, 9.31943127461857347042340519888, 10.26549697111277613913854317503, 10.48883831864066842202469277862, 12.060857861472021479754520782905, 12.87720043102636093558609592012, 14.209524160449302617849093767965, 15.30531739284018272872247694621, 16.01055591732374830273296242923, 16.80297332238666911221099080038, 17.45340709941177532248415237622, 18.05870947532728526376031589412, 19.42891893478635039627280576618, 20.06994425420810321462427541434, 21.16531970019683961285613938137, 21.93647397538546456820508460116, 22.94203603530130342507673303110, 23.75100361896263415102890980358, 24.78302461309676000271738315865

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