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Properties

Label 1-407-407.316-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: $\chi_{407} (316, \cdot )$
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

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

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