↑

Properties

Label 1-407-407.379-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.988 + 0.149i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.559 − 0.829i)2-s + (0.0348 − 0.999i)3-s + (−0.374 − 0.927i)4-s + (0.438 − 0.898i)5-s + (−0.809 − 0.587i)6-s + (0.990 + 0.139i)7-s + (−0.978 − 0.207i)8-s + (−0.997 − 0.0697i)9-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.241 + 0.970i)13-s + (0.669 − 0.743i)14-s + (−0.882 − 0.469i)15-s + (−0.719 + 0.694i)16-s + (−0.241 − 0.970i)17-s + (−0.615 + 0.788i)18-s + ⋯
L(s)  = 1  + (0.559 − 0.829i)2-s + (0.0348 − 0.999i)3-s + (−0.374 − 0.927i)4-s + (0.438 − 0.898i)5-s + (−0.809 − 0.587i)6-s + (0.990 + 0.139i)7-s + (−0.978 − 0.207i)8-s + (−0.997 − 0.0697i)9-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.241 + 0.970i)13-s + (0.669 − 0.743i)14-s + (−0.882 − 0.469i)15-s + (−0.719 + 0.694i)16-s + (−0.241 − 0.970i)17-s + (−0.615 + 0.788i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.988 + 0.149i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.988 + 0.149i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1316218208 - 1.755486184i\)
\(L(\frac12)\) \(\approx\) \(-0.1316218208 - 1.755486184i\)
\(L(1)\) \(\approx\) \(0.7414190668 - 1.230823450i\)
\(L(1)\) \(\approx\) \(0.7414190668 - 1.230823450i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.559 - 0.829i)T \)
3 \( 1 + (0.0348 - 0.999i)T \)
5 \( 1 + (0.438 - 0.898i)T \)
7 \( 1 + (0.990 + 0.139i)T \)
13 \( 1 + (-0.241 + 0.970i)T \)
17 \( 1 + (-0.241 - 0.970i)T \)
19 \( 1 + (0.0348 - 0.999i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.848 + 0.529i)T \)
43 \( 1 + T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (0.438 + 0.898i)T \)
59 \( 1 + (-0.882 - 0.469i)T \)
61 \( 1 + (-0.997 + 0.0697i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.559 + 0.829i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.719 - 0.694i)T \)
83 \( 1 + (-0.241 - 0.970i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (0.913 + 0.406i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−24.863829284784606322717870162196, −23.848237974944593774741664354515, −22.84060291620200016599171801558, −22.32031256171022962705511593737, −21.377507481527413140315367563312, −20.97776345577032269218531820326, −19.76752017963301841623519056724, −18.28295526568964064541993342916, −17.48452081291183674662953930602, −16.980404106487632076394040390281, −15.652677805504898702785150933738, −15.09634163054946587149016021319, −14.3665001988014174154379704379, −13.75950677063874851601638589241, −12.405359935411620331216811824942, −11.235574233000893387594398238834, −10.45901071571233808578629181141, −9.43556951337049761349978723026, −8.178747084619858489343573451860, −7.56151666191321628489751757109, −6.006037255212673769122885720590, −5.552980435830977538725205762644, −4.300411631796742328620154997347, −3.506734149739600608727851695784, −2.29588320386611269839674952834, 0.869332920249461964820596250802, 1.87679443563101764366534756404, 2.642363763921997480300685273199, 4.44116034576176951737988346352, 5.093857665469148773750857838903, 6.16011103099231489853148188418, 7.322675801000135446202290373555, 8.75656621683758307415146645473, 9.16163567896597418239976650402, 10.70621426126808609309894977441, 11.67197981734263886358734793907, 12.23544540032829732728513981517, 13.12912372202455376989670900263, 14.03140276583171372425615552533, 14.41952488155091177824227719190, 15.92779456167484507492110325253, 17.17366195401349150504547935405, 18.022586387571187410703635576787, 18.606849281793747400800035733009, 19.87755592710516903918441372800, 20.240677839325832283303818045604, 21.282080976821051606217275473312, 21.92724250666742378898520035541, 23.161481029144678011757370126527, 24.01134168543430639684138471399

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