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Properties

Label 1-407-407.193-r0-0-0
Degree $1$
Conductor $407$
Sign $0.954 - 0.299i$
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  + (0.994 − 0.104i)2-s + (−0.669 + 0.743i)3-s + (0.978 − 0.207i)4-s + (−0.994 − 0.104i)5-s + (−0.587 + 0.809i)6-s + (0.978 − 0.207i)7-s + (0.951 − 0.309i)8-s + (−0.104 − 0.994i)9-s − 10-s + (−0.5 + 0.866i)12-s + (0.406 − 0.913i)13-s + (0.951 − 0.309i)14-s + (0.743 − 0.669i)15-s + (0.913 − 0.406i)16-s + (−0.406 − 0.913i)17-s + (−0.207 − 0.978i)18-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)2-s + (−0.669 + 0.743i)3-s + (0.978 − 0.207i)4-s + (−0.994 − 0.104i)5-s + (−0.587 + 0.809i)6-s + (0.978 − 0.207i)7-s + (0.951 − 0.309i)8-s + (−0.104 − 0.994i)9-s − 10-s + (−0.5 + 0.866i)12-s + (0.406 − 0.913i)13-s + (0.951 − 0.309i)14-s + (0.743 − 0.669i)15-s + (0.913 − 0.406i)16-s + (−0.406 − 0.913i)17-s + (−0.207 − 0.978i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.954 - 0.299i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.954 - 0.299i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.866215467 - 0.2861958631i\)
\(L(\frac12)\) \(\approx\) \(1.866215467 - 0.2861958631i\)
\(L(1)\) \(\approx\) \(1.515846021 - 0.03947181134i\)
\(L(1)\) \(\approx\) \(1.515846021 - 0.03947181134i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.994 - 0.104i)T \)
3 \( 1 + (-0.669 + 0.743i)T \)
5 \( 1 + (-0.994 - 0.104i)T \)
7 \( 1 + (0.978 - 0.207i)T \)
13 \( 1 + (0.406 - 0.913i)T \)
17 \( 1 + (-0.406 - 0.913i)T \)
19 \( 1 + (-0.743 - 0.669i)T \)
23 \( 1 + iT \)
29 \( 1 + (0.951 + 0.309i)T \)
31 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.309 + 0.951i)T \)
53 \( 1 + (-0.104 - 0.994i)T \)
59 \( 1 + (-0.743 + 0.669i)T \)
61 \( 1 + (-0.994 - 0.104i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.104 + 0.994i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.406 - 0.913i)T \)
83 \( 1 + (-0.913 + 0.406i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.587 - 0.809i)T \)
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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

−24.369840141875623873281598055865, −23.25904851523461621585355782053, −23.12267030632994455307695672261, −21.851113085741917058326125152957, −21.176096830635127678827242036669, −20.05354614154993396866907089502, −19.16524758132476328892619072778, −18.39470059879790798081063147210, −17.136307585421964216100667834549, −16.47474746160094021266140230490, −15.42138927964424453117609336020, −14.59948631372471832510083474515, −13.741870699965076707345448537037, −12.58613584044712367075957911, −12.03368793398489406592945956619, −11.1900044095896848366173999166, −10.671875653387655662988914775477, −8.35995651923853221919679674959, −7.87940028863032937372623314462, −6.67409458155327665619050943750, −6.05619937196207541803553249942, −4.65061779379287618759733262970, −4.18021206687042297266939876294, −2.50866042847061361853789588700, −1.44068147947009023608479914733, 0.999254229260926434806555512941, 2.87305712185135337690861369651, 3.93544708022444237983101192171, 4.72691831194095650578071022412, 5.37918244849693509154474224267, 6.68380082750310581755360426772, 7.6636622151753446934881816014, 8.83381329834230289250553863241, 10.44331795569392467160112751774, 11.02157983288515522318577769836, 11.72620371125750988368015355313, 12.49310732391630540605235711090, 13.719853164524276222955762582299, 14.744364004505376707421828622483, 15.597744409928342337357047520190, 15.894372885098191069898949090940, 17.15408274363446475510301727852, 17.95899386970043559085023636863, 19.42970149983095827535381119314, 20.29326022643050403393018487622, 20.91360854192537771184863138445, 21.75360342697233926789295695850, 22.67381798791093208181306819901, 23.344783074726747075178571097, 23.869532072663828159662527583872

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