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Properties

Label 1-407-407.161-r0-0-0
Degree $1$
Conductor $407$
Sign $0.273 - 0.961i$
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  + (0.999 + 0.0348i)2-s + (−0.961 − 0.275i)3-s + (0.997 + 0.0697i)4-s + (0.469 − 0.882i)5-s + (−0.951 − 0.309i)6-s + (−0.438 − 0.898i)7-s + (0.994 + 0.104i)8-s + (0.848 + 0.529i)9-s + (0.5 − 0.866i)10-s + (−0.939 − 0.342i)12-s + (0.927 + 0.374i)13-s + (−0.406 − 0.913i)14-s + (−0.694 + 0.719i)15-s + (0.990 + 0.139i)16-s + (−0.927 + 0.374i)17-s + (0.829 + 0.559i)18-s + ⋯
L(s)  = 1  + (0.999 + 0.0348i)2-s + (−0.961 − 0.275i)3-s + (0.997 + 0.0697i)4-s + (0.469 − 0.882i)5-s + (−0.951 − 0.309i)6-s + (−0.438 − 0.898i)7-s + (0.994 + 0.104i)8-s + (0.848 + 0.529i)9-s + (0.5 − 0.866i)10-s + (−0.939 − 0.342i)12-s + (0.927 + 0.374i)13-s + (−0.406 − 0.913i)14-s + (−0.694 + 0.719i)15-s + (0.990 + 0.139i)16-s + (−0.927 + 0.374i)17-s + (0.829 + 0.559i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.273 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.273 - 0.961i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.273 - 0.961i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.553068443 - 1.173511600i\)
\(L(\frac12)\) \(\approx\) \(1.553068443 - 1.173511600i\)
\(L(1)\) \(\approx\) \(1.468076143 - 0.5152464407i\)
\(L(1)\) \(\approx\) \(1.468076143 - 0.5152464407i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.999 + 0.0348i)T \)
3 \( 1 + (-0.961 - 0.275i)T \)
5 \( 1 + (0.469 - 0.882i)T \)
7 \( 1 + (-0.438 - 0.898i)T \)
13 \( 1 + (0.927 + 0.374i)T \)
17 \( 1 + (-0.927 + 0.374i)T \)
19 \( 1 + (0.275 - 0.961i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.406 + 0.913i)T \)
31 \( 1 + (0.951 - 0.309i)T \)
41 \( 1 + (-0.241 - 0.970i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (-0.882 + 0.469i)T \)
59 \( 1 + (0.694 - 0.719i)T \)
61 \( 1 + (0.529 + 0.848i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (0.0348 + 0.999i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.139 - 0.990i)T \)
83 \( 1 + (0.374 + 0.927i)T \)
89 \( 1 + (0.984 - 0.173i)T \)
97 \( 1 + (0.207 - 0.978i)T \)
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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

−24.42306159096485693693758262615, −23.214697879153447007763863719271, −22.6917080671283655499899327793, −22.095853080971159078730174404046, −21.39076709448888807949988063358, −20.591035502004142813682130346402, −19.20929736979155726514259519123, −18.324690466914915118543806093721, −17.56428698320885775525586317607, −16.25938530489185130715957186876, −15.65332825868857216850943629833, −14.94861749832879075933328786921, −13.76582896026309766877151334272, −12.98128943700293261817486225025, −11.912868298482284761852522714812, −11.30617877346503018549846350534, −10.37182822719904422958476484823, −9.54810339540999073715768936337, −7.79937375916432240311988557719, −6.34722697335982406005915654175, −6.209181530699497505750159184191, −5.22627710227509191776131551215, −3.95125506011713360431539311472, −2.95591270639357420875470005049, −1.73632875478252462643916332213, 0.97282641714117138463116558507, 2.12430929513564776031978146677, 3.91571471706121485844089195348, 4.59299767961631746397729642956, 5.65018242695484982803613952257, 6.46553044770007746960382992607, 7.218556751474279784268888345656, 8.63608049536102694557372911556, 10.1009076257873515539356686386, 10.88990073991121149602342881370, 11.80227710303217945896582871344, 12.77745952313200375489180837353, 13.37302935766708042723736954558, 13.97186978103418676399297309456, 15.676415735033864494618588412303, 16.147159647635701966789510644246, 17.03031303555168451949771712655, 17.66928504180435401342494609264, 19.057595668679615518111918666840, 20.16447844836940539554742346255, 20.71420723604798777663706872537, 21.912910352854398035920035094351, 22.30504911389203221074661589484, 23.549065147927737504300633432042, 23.82729723140710644760165273622

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