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Properties

Label 1-407-407.140-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.741 + 0.671i$
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  + (0.743 − 0.669i)2-s + (−0.913 − 0.406i)3-s + (0.104 − 0.994i)4-s + (−0.743 − 0.669i)5-s + (−0.951 + 0.309i)6-s + (0.104 − 0.994i)7-s + (−0.587 − 0.809i)8-s + (0.669 + 0.743i)9-s − 10-s + (−0.5 + 0.866i)12-s + (−0.207 − 0.978i)13-s + (−0.587 − 0.809i)14-s + (0.406 + 0.913i)15-s + (−0.978 − 0.207i)16-s + (0.207 − 0.978i)17-s + (0.994 + 0.104i)18-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (−0.913 − 0.406i)3-s + (0.104 − 0.994i)4-s + (−0.743 − 0.669i)5-s + (−0.951 + 0.309i)6-s + (0.104 − 0.994i)7-s + (−0.587 − 0.809i)8-s + (0.669 + 0.743i)9-s − 10-s + (−0.5 + 0.866i)12-s + (−0.207 − 0.978i)13-s + (−0.587 − 0.809i)14-s + (0.406 + 0.913i)15-s + (−0.978 − 0.207i)16-s + (0.207 − 0.978i)17-s + (0.994 + 0.104i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.741 + 0.671i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.741 + 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3278542238 - 0.8501949332i\)
\(L(\frac12)\) \(\approx\) \(-0.3278542238 - 0.8501949332i\)
\(L(1)\) \(\approx\) \(0.5354763965 - 0.7701388777i\)
\(L(1)\) \(\approx\) \(0.5354763965 - 0.7701388777i\)

Euler product

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

−24.420165103589169109144272292630, −23.933691828663619424615639362698, −23.07802673985589677983514705047, −22.3895918773456204000472129780, −21.5538066376742179598549037352, −21.21397895625900062849618199032, −19.52785331643474275404836408736, −18.6199755440447757971513667979, −17.62393545961515421694800259743, −16.86314029008351113913162025407, −15.80232512579953036781944196740, −15.290023473446944914839041952797, −14.66127982693902850755110843021, −13.332780780959750988609639162758, −12.16790833936949194570336869807, −11.69980319147402892350889077834, −10.913951055646127198339824337600, −9.4671729986038590459470268010, −8.34892830917338321037595189135, −7.18326216827194970511190614091, −6.37454991805653952735080468446, −5.54314536110461252140117846987, −4.4503869118161054260666699231, −3.67968396576668001853357151631, −2.31777765882822599211916674661, 0.48940430696513872865562006676, 1.42733489660691986839015851234, 3.11601363700412130351246278348, 4.37236746364329697113452383064, 4.92607999850671785607217464933, 6.03964847037407019455258626090, 7.1692272385792825850133989891, 8.080767269213853513330163418370, 9.74075885289080883438559546131, 10.66091764649045757575792413544, 11.32306001408950699006533792756, 12.379927245553366370834432367373, 12.76283281376318146870235044802, 13.7748950150441981339564393908, 14.82486612672624664706530494460, 16.03977123557477960592370090576, 16.60736143088559228106034500301, 17.75667695666678431027941470452, 18.71984849297217749118573179992, 19.632755918519597874851759371562, 20.395234614626048921469759221518, 21.07260364166739655403145977592, 22.389720439412345477037093086053, 23.05402681141366745694634763221, 23.38448745877435051899519000742

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