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Properties

Label 1-407-407.205-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.211 + 0.977i$
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  + (−0.788 + 0.615i)2-s + (−0.559 − 0.829i)3-s + (0.241 − 0.970i)4-s + (0.139 − 0.990i)5-s + (0.951 + 0.309i)6-s + (0.719 + 0.694i)7-s + (0.406 + 0.913i)8-s + (−0.374 + 0.927i)9-s + (0.5 + 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.529 + 0.848i)13-s + (−0.994 − 0.104i)14-s + (−0.898 + 0.438i)15-s + (−0.882 − 0.469i)16-s + (0.529 + 0.848i)17-s + (−0.275 − 0.961i)18-s + ⋯
L(s)  = 1  + (−0.788 + 0.615i)2-s + (−0.559 − 0.829i)3-s + (0.241 − 0.970i)4-s + (0.139 − 0.990i)5-s + (0.951 + 0.309i)6-s + (0.719 + 0.694i)7-s + (0.406 + 0.913i)8-s + (−0.374 + 0.927i)9-s + (0.5 + 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.529 + 0.848i)13-s + (−0.994 − 0.104i)14-s + (−0.898 + 0.438i)15-s + (−0.882 − 0.469i)16-s + (0.529 + 0.848i)17-s + (−0.275 − 0.961i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.211 + 0.977i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.211 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2614151468 + 0.3238714987i\)
\(L(\frac12)\) \(\approx\) \(0.2614151468 + 0.3238714987i\)
\(L(1)\) \(\approx\) \(0.5475528593 + 0.06783401805i\)
\(L(1)\) \(\approx\) \(0.5475528593 + 0.06783401805i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.788 + 0.615i)T \)
3 \( 1 + (-0.559 - 0.829i)T \)
5 \( 1 + (0.139 - 0.990i)T \)
7 \( 1 + (0.719 + 0.694i)T \)
13 \( 1 + (-0.529 + 0.848i)T \)
17 \( 1 + (0.529 + 0.848i)T \)
19 \( 1 + (-0.829 + 0.559i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (-0.994 + 0.104i)T \)
31 \( 1 + (-0.951 + 0.309i)T \)
41 \( 1 + (-0.997 + 0.0697i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.104 + 0.994i)T \)
53 \( 1 + (0.990 - 0.139i)T \)
59 \( 1 + (0.898 - 0.438i)T \)
61 \( 1 + (-0.927 + 0.374i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.615 + 0.788i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.469 - 0.882i)T \)
83 \( 1 + (-0.848 + 0.529i)T \)
89 \( 1 + (0.984 + 0.173i)T \)
97 \( 1 + (-0.743 - 0.669i)T \)
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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

−23.96371404303348432525439844314, −22.864468275875368008794453909250, −22.186042804133569198817785590430, −21.47150370509941883738363553210, −20.53193218700242497751387699358, −19.96154510703471800068259934924, −18.625876195780781299315435279584, −17.95815110611284780396184794393, −17.21182274564538566704364450313, −16.517659218152582737315659834071, −15.31366969693193169220523440457, −14.57558737218932360164804078764, −13.34715409887047551208642490797, −11.9878095405300877953779893664, −11.299293355699373525465669971776, −10.398503103156873281386643411975, −10.12394220507234783980801816609, −8.892316334957427796778274474112, −7.65934391992606740327797578730, −6.88879250332322357665205417203, −5.4939979865469555371896025704, −4.21485060173583222563409070553, −3.30727925364543587290644002985, −2.12478104739679947603561857386, −0.33095865274858435807415249823, 1.548527761266182633525517509612, 1.9414261532983332258659906193, 4.50917499390640698403578670529, 5.53161971717709718601331671269, 6.08391015928158053981235097132, 7.39354335930357195342653962138, 8.18888166266464940743476838070, 8.8967812263839403377286855967, 10.02152336480440796411390164033, 11.24411999493713436520911019402, 12.02215438695146053078688242400, 12.8975203634919621643644545052, 14.13024357444689515468332788562, 14.91140472834354515476623491213, 16.20831265611336042007551159833, 16.844675882781749957382336887570, 17.4921857085297386412093832767, 18.35231592574393470965360546277, 19.12213302238172918540314825986, 19.8986441881430590856777655782, 21.065047158976772544595630433792, 21.9795599866025812305008850610, 23.41549248186428862737102529696, 23.954071922949296572375158126999, 24.47835447423043122706414470461

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