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Properties

Label 1-407-407.167-r0-0-0
Degree $1$
Conductor $407$
Sign $0.883 + 0.468i$
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  + (0.898 + 0.438i)2-s + (0.882 + 0.469i)3-s + (0.615 + 0.788i)4-s + (0.0697 − 0.997i)5-s + (0.587 + 0.809i)6-s + (0.374 − 0.927i)7-s + (0.207 + 0.978i)8-s + (0.559 + 0.829i)9-s + (0.5 − 0.866i)10-s + (0.173 + 0.984i)12-s + (0.275 − 0.961i)13-s + (0.743 − 0.669i)14-s + (0.529 − 0.848i)15-s + (−0.241 + 0.970i)16-s + (−0.275 − 0.961i)17-s + (0.139 + 0.990i)18-s + ⋯
L(s)  = 1  + (0.898 + 0.438i)2-s + (0.882 + 0.469i)3-s + (0.615 + 0.788i)4-s + (0.0697 − 0.997i)5-s + (0.587 + 0.809i)6-s + (0.374 − 0.927i)7-s + (0.207 + 0.978i)8-s + (0.559 + 0.829i)9-s + (0.5 − 0.866i)10-s + (0.173 + 0.984i)12-s + (0.275 − 0.961i)13-s + (0.743 − 0.669i)14-s + (0.529 − 0.848i)15-s + (−0.241 + 0.970i)16-s + (−0.275 − 0.961i)17-s + (0.139 + 0.990i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 + 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.883 + 0.468i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.883 + 0.468i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.020046451 + 0.7506198769i\)
\(L(\frac12)\) \(\approx\) \(3.020046451 + 0.7506198769i\)
\(L(1)\) \(\approx\) \(2.256537395 + 0.4896460017i\)
\(L(1)\) \(\approx\) \(2.256537395 + 0.4896460017i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.898 + 0.438i)T \)
3 \( 1 + (0.882 + 0.469i)T \)
5 \( 1 + (0.0697 - 0.997i)T \)
7 \( 1 + (0.374 - 0.927i)T \)
13 \( 1 + (0.275 - 0.961i)T \)
17 \( 1 + (-0.275 - 0.961i)T \)
19 \( 1 + (-0.469 + 0.882i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (0.743 + 0.669i)T \)
31 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (0.0348 + 0.999i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.669 + 0.743i)T \)
53 \( 1 + (-0.997 + 0.0697i)T \)
59 \( 1 + (-0.529 + 0.848i)T \)
61 \( 1 + (0.829 + 0.559i)T \)
67 \( 1 + (-0.766 - 0.642i)T \)
71 \( 1 + (0.438 + 0.898i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.970 + 0.241i)T \)
83 \( 1 + (-0.961 + 0.275i)T \)
89 \( 1 + (-0.642 - 0.766i)T \)
97 \( 1 + (0.406 + 0.913i)T \)
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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

−23.98481370622133031801490402746, −23.64867328893011529535501045057, −22.27388854134813454042834378574, −21.61229688981330729278928742657, −21.06081560028856727432305245920, −19.833967390354820307221578234111, −19.13755192557534927127248860803, −18.57364275942618725961920930912, −17.58094265246321000737572904750, −15.73026488145966698625469443409, −15.18241071032970194783909134554, −14.40337203353240779587711125284, −13.75686208980920539312763536569, −12.81498831763238967305235808647, −11.8207794964397265669837251954, −11.093753461775413225609458283427, −9.903109985808086397297275662956, −8.92208962133450428370585745022, −7.71111301835372705860163165424, −6.58373605719099596234401930818, −5.98934480546843438967703269850, −4.39118599983878749645005493230, −3.46090357471029140727893881017, −2.32303731581504739783790237799, −1.86908784376773649421796452734, 1.554122189458865401116399691152, 2.936000388202437462464254842565, 4.03169715213998331924474867766, 4.66937307952595816495139443017, 5.64292415299685551071016257614, 7.12194920178135050798288389537, 8.05509829005019619692910058643, 8.629643280527623241291006039793, 10.01147031517388892039649362397, 10.93213861207975456176910887195, 12.30240432486757793216445159515, 13.06536059334833891563464452488, 13.92231804136885195134707875569, 14.45531384878881548154995374859, 15.663993411273810664874988036260, 16.21726244546943670500866533637, 17.02714333639779717729953924160, 18.10509006739913735285380126579, 19.823131614853474901886392690247, 20.303478585759240366149101969742, 20.822160936888060697682593201575, 21.6820016832980001909510369865, 22.73007765475493332782583774174, 23.62328855302362377599712371546, 24.43146302780272709837873792982

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