Properties

Label 1-407-407.304-r0-0-0
Degree $1$
Conductor $407$
Sign $0.0816 + 0.996i$
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  + (0.207 − 0.978i)2-s + (0.104 − 0.994i)3-s + (−0.913 − 0.406i)4-s + (−0.207 − 0.978i)5-s + (−0.951 − 0.309i)6-s + (−0.913 − 0.406i)7-s + (−0.587 + 0.809i)8-s + (−0.978 − 0.207i)9-s − 10-s + (−0.5 + 0.866i)12-s + (−0.743 − 0.669i)13-s + (−0.587 + 0.809i)14-s + (−0.994 + 0.104i)15-s + (0.669 + 0.743i)16-s + (0.743 − 0.669i)17-s + (−0.406 + 0.913i)18-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)2-s + (0.104 − 0.994i)3-s + (−0.913 − 0.406i)4-s + (−0.207 − 0.978i)5-s + (−0.951 − 0.309i)6-s + (−0.913 − 0.406i)7-s + (−0.587 + 0.809i)8-s + (−0.978 − 0.207i)9-s − 10-s + (−0.5 + 0.866i)12-s + (−0.743 − 0.669i)13-s + (−0.587 + 0.809i)14-s + (−0.994 + 0.104i)15-s + (0.669 + 0.743i)16-s + (0.743 − 0.669i)17-s + (−0.406 + 0.913i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0816 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0816 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.0816 + 0.996i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.0816 + 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.5070610382 - 0.4672380497i\)
\(L(\frac12)\) \(\approx\) \(-0.5070610382 - 0.4672380497i\)
\(L(1)\) \(\approx\) \(0.2941091457 - 0.7523676607i\)
\(L(1)\) \(\approx\) \(0.2941091457 - 0.7523676607i\)

Euler product

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

−25.2043884982713260476801370122, −24.10703331314063861814683996200, −23.01445350508033878645975185521, −22.38342572601535204737636012848, −21.94376243028245237496160119027, −21.00580133257531557422543608393, −19.5478388400812947840470621617, −18.89429899305561271884065334912, −17.85100011847432556967796523096, −16.74361092506851885380577309903, −16.13269430557173197669012430852, −15.32608210381684218186641637585, −14.55560527513737896862237409485, −14.01058321217456551897051563978, −12.62969449322640781486223044697, −11.651317360822218789288311735898, −10.27254534771081367184993651170, −9.66538049599787597829099503218, −8.71544686514544012755771082435, −7.55811766650685364442436560679, −6.56387560633462253942511872371, −5.72448307588860703874534217960, −4.60039077890974760482958642594, −3.51462717849058979519730877925, −2.814257767520948170946738596964, 0.391267510139606786373155505884, 1.377869491595962129717603092342, 2.76847964435441651290957729165, 3.62625858283583764494469195925, 5.05864540761318435756139598497, 5.85721055250510872788666955190, 7.36965188126179594366820056937, 8.179281874554879355324578515343, 9.43661306327519368681915062535, 9.9482888794450306467473282810, 11.60462437090236259588571540403, 12.03339425632700045688895362391, 13.06681318129042571513567901050, 13.38109388055987007008176407853, 14.42693705689393293727121432138, 15.76024504909414901664242069541, 16.977362807280948304955823406060, 17.63122427630114622880286307381, 18.804765209154536806452305658284, 19.41148530048354531607310976131, 20.199590440511757792065720879664, 20.64549100547247889309000572170, 22.01477681630385919985149373639, 22.94221123884007104141389516421, 23.41499160955927034123401049578

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