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Properties

Label 1-407-407.208-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.881 - 0.471i$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)5-s − i·6-s + (0.5 − 0.866i)7-s − i·8-s + (−0.5 − 0.866i)9-s − 10-s + (−0.5 − 0.866i)12-s + (0.866 + 0.5i)13-s − i·14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 − 0.5i)18-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)3-s + (0.5 − 0.866i)4-s + (−0.866 − 0.5i)5-s − i·6-s + (0.5 − 0.866i)7-s − i·8-s + (−0.5 − 0.866i)9-s − 10-s + (−0.5 − 0.866i)12-s + (0.866 + 0.5i)13-s − i·14-s + (−0.866 + 0.5i)15-s + (−0.5 − 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 − 0.5i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.881 - 0.471i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (208, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5542336229 - 2.209802310i\)
\(L(\frac12)\) \(\approx\) \(0.5542336229 - 2.209802310i\)
\(L(1)\) \(\approx\) \(1.204960173 - 1.319426798i\)
\(L(1)\) \(\approx\) \(1.204960173 - 1.319426798i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.866 + 0.5i)T \)
17 \( 1 + (-0.866 + 0.5i)T \)
19 \( 1 + (0.866 + 0.5i)T \)
23 \( 1 + iT \)
29 \( 1 + iT \)
31 \( 1 - iT \)
41 \( 1 + (-0.5 + 0.866i)T \)
43 \( 1 - iT \)
47 \( 1 + T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (-0.866 - 0.5i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.866 + 0.5i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + iT \)
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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

−24.739077855748137875444467071039, −23.81056589890997847838466126988, −22.598196337757471508908092400018, −22.38370593921346428561012339121, −21.33407315647463480681859098233, −20.546295930419811885816980054129, −19.84002915094883397442584366343, −18.56954477759273952230274817167, −17.61881405979475853885663831891, −16.21114024401383181442953029614, −15.65862010753593456173624142998, −15.16584071075331053327776440464, −14.29180508870600174757928918181, −13.47468170096122780090585058923, −12.16399177792153216359020711036, −11.343884597870422897899601259500, −10.645683245813116980794218265058, −8.96721982300478578926171916291, −8.31941816839782639757372654304, −7.39413414581946256623362037217, −6.13762517604370979603267878075, −5.02348432572161874341374471929, −4.2509152277509523909892235791, −3.17755030467273816398611663006, −2.46083159746512520886835849026, 1.01605483865048818837912611932, 1.85284921167311170301857431103, 3.47716532545486881439242385616, 3.992047547316998626926591748030, 5.24594050194190763796882658068, 6.54770756269937308155228595468, 7.41205144626332324028042385692, 8.33489880504778538587583063086, 9.47760422583055839053114538204, 10.9871082328787198136808224733, 11.53994219012512677474077240304, 12.47366860167002122960403628468, 13.383779323793380122005322463505, 13.9071593802840346332995527622, 14.92475721644483353368109927724, 15.78333714746089800246297408791, 16.88440331469886663164894789171, 18.156745104228035382987742184160, 19.01855493870892804935298522204, 19.91988084509602818885497810352, 20.332971544265646162846326763772, 21.08612079666571291817963886847, 22.39674841590850265861514535608, 23.464716402233680780155586761907, 23.76045078367007056711344334413

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