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Properties

Label 1-407-407.131-r0-0-0
Degree $1$
Conductor $407$
Sign $0.0320 + 0.999i$
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  + (0.342 + 0.939i)2-s + (0.939 + 0.342i)3-s + (−0.766 + 0.642i)4-s + (0.984 − 0.173i)5-s + i·6-s + (−0.173 − 0.984i)7-s + (−0.866 − 0.5i)8-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)10-s + (−0.939 + 0.342i)12-s + (0.642 + 0.766i)13-s + (0.866 − 0.5i)14-s + (0.984 + 0.173i)15-s + (0.173 − 0.984i)16-s + (−0.642 + 0.766i)17-s + (−0.342 + 0.939i)18-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (0.939 + 0.342i)3-s + (−0.766 + 0.642i)4-s + (0.984 − 0.173i)5-s + i·6-s + (−0.173 − 0.984i)7-s + (−0.866 − 0.5i)8-s + (0.766 + 0.642i)9-s + (0.5 + 0.866i)10-s + (−0.939 + 0.342i)12-s + (0.642 + 0.766i)13-s + (0.866 − 0.5i)14-s + (0.984 + 0.173i)15-s + (0.173 − 0.984i)16-s + (−0.642 + 0.766i)17-s + (−0.342 + 0.939i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0320 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0320 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.0320 + 0.999i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.0320 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.659245285 + 1.606966880i\)
\(L(\frac12)\) \(\approx\) \(1.659245285 + 1.606966880i\)
\(L(1)\) \(\approx\) \(1.495516720 + 0.9359526489i\)
\(L(1)\) \(\approx\) \(1.495516720 + 0.9359526489i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.342 + 0.939i)T \)
3 \( 1 + (0.939 + 0.342i)T \)
5 \( 1 + (0.984 - 0.173i)T \)
7 \( 1 + (-0.173 - 0.984i)T \)
13 \( 1 + (0.642 + 0.766i)T \)
17 \( 1 + (-0.642 + 0.766i)T \)
19 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.173 - 0.984i)T \)
59 \( 1 + (-0.984 - 0.173i)T \)
61 \( 1 + (-0.642 - 0.766i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + T \)
79 \( 1 + (-0.984 + 0.173i)T \)
83 \( 1 + (-0.766 - 0.642i)T \)
89 \( 1 + (0.984 + 0.173i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−24.35443589690644871996487962349, −22.92790429623162733723568191397, −22.22411891543069736824152672919, −21.33833390880000092646280878732, −20.66737064793356484199574010321, −19.98996789587760881165220963678, −18.80456563863808592186848343359, −18.34081405956820268612428866891, −17.68650683187749162524727558056, −15.90125840001548160504325978621, −14.97351724070553104675421962521, −14.11209271799421587574285401693, −13.446378987040789995041532241966, −12.67153558355499700325919741433, −11.82270168720453187984278939265, −10.48430290179841168686019298299, −9.66045342666854218289677442738, −8.96654735652589612154159591350, −8.02877564158730771982089667361, −6.32670527032196571068711655968, −5.65123854216361170447131130698, −4.249503359641597511837893913232, −2.92517912506511953434404937709, −2.43198034496025983740719774648, −1.32791669831735184786562337790, 1.60034886893711397381841117382, 3.101296968992245072467701712244, 4.12711136853701563377151360374, 4.935640095985710310634598821840, 6.32820908554842967538279128886, 7.037520768098591179303727118218, 8.22123911033595218525513319975, 9.037848336505201780435924412933, 9.79896230972290209912612292676, 10.83932418610605423866108101558, 12.60868799734358165748856520546, 13.585667459296769095731000764003, 13.79798823916355505228465703808, 14.67758943019992965982191896201, 15.8466501343745502018941471311, 16.39630887602573623724023589579, 17.460711769158743568141091745222, 18.129953875584580908740195602723, 19.46363685739680539770801628122, 20.281998021999752458665766283441, 21.518018604007606343092382050, 21.597000016565001228674604478917, 22.90024519140648525819220354431, 23.96762797752924235320760583170, 24.496843760334162158332355344960

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