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Properties

Label 1-407-407.81-r0-0-0
Degree $1$
Conductor $407$
Sign $0.787 + 0.616i$
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  + (0.848 + 0.529i)2-s + (−0.241 − 0.970i)3-s + (0.438 + 0.898i)4-s + (0.0348 + 0.999i)5-s + (0.309 − 0.951i)6-s + (0.559 − 0.829i)7-s + (−0.104 + 0.994i)8-s + (−0.882 + 0.469i)9-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (0.990 − 0.139i)13-s + (0.913 − 0.406i)14-s + (0.961 − 0.275i)15-s + (−0.615 + 0.788i)16-s + (0.990 + 0.139i)17-s + (−0.997 − 0.0697i)18-s + ⋯
L(s)  = 1  + (0.848 + 0.529i)2-s + (−0.241 − 0.970i)3-s + (0.438 + 0.898i)4-s + (0.0348 + 0.999i)5-s + (0.309 − 0.951i)6-s + (0.559 − 0.829i)7-s + (−0.104 + 0.994i)8-s + (−0.882 + 0.469i)9-s + (−0.5 + 0.866i)10-s + (0.766 − 0.642i)12-s + (0.990 − 0.139i)13-s + (0.913 − 0.406i)14-s + (0.961 − 0.275i)15-s + (−0.615 + 0.788i)16-s + (0.990 + 0.139i)17-s + (−0.997 − 0.0697i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $0.787 + 0.616i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ 0.787 + 0.616i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.022557308 + 0.6974791016i\)
\(L(\frac12)\) \(\approx\) \(2.022557308 + 0.6974791016i\)
\(L(1)\) \(\approx\) \(1.633832660 + 0.3433565907i\)
\(L(1)\) \(\approx\) \(1.633832660 + 0.3433565907i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.848 + 0.529i)T \)
3 \( 1 + (-0.241 - 0.970i)T \)
5 \( 1 + (0.0348 + 0.999i)T \)
7 \( 1 + (0.559 - 0.829i)T \)
13 \( 1 + (0.990 - 0.139i)T \)
17 \( 1 + (0.990 + 0.139i)T \)
19 \( 1 + (-0.241 - 0.970i)T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.913 + 0.406i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.719 + 0.694i)T \)
43 \( 1 + T \)
47 \( 1 + (0.913 - 0.406i)T \)
53 \( 1 + (0.0348 - 0.999i)T \)
59 \( 1 + (0.961 - 0.275i)T \)
61 \( 1 + (-0.882 - 0.469i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.848 - 0.529i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.615 - 0.788i)T \)
83 \( 1 + (0.990 + 0.139i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (-0.978 - 0.207i)T \)
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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

−24.00708222563770209509538296292, −23.23681566093119104984283390637, −22.39118851969570684763902651658, −21.44915961859485029578350460213, −20.767189901474525105967861207961, −20.58792049174288014546902925126, −19.20684757614884944434302904553, −18.25575237534168198955844291630, −16.9468093698221981695992325524, −16.095685723654598970537748040955, −15.450793657097287321506641604850, −14.481732028321515787383278730979, −13.685811645603118231668873430525, −12.25650643695917847316004501345, −12.01741123561400456520846185717, −10.882154797609560380166103535289, −9.984599465546543281798178477003, −9.015227990643504149053279821868, −8.13048582070929855020574962107, −6.01109889396547339084563684309, −5.6396855177928601765312107584, −4.547478086044181712344819560905, −3.8900697047954291716401641318, −2.536106246112729849968950999932, −1.17416564476985235855024516806, 1.46414600669464771779490443004, 2.81590677914170097560730772275, 3.754182632718285847972964362429, 5.138742378000274660395656195040, 6.16350285843091192612756087491, 6.9269742148340117858398598037, 7.6456971534305834968449051878, 8.47917743918748907064962086047, 10.450926508749042552109595893494, 11.229434744295253770165542154249, 11.97947850379538662329596390144, 13.16329870106593552124702041293, 13.875646950110393658371949192886, 14.37233057635729524631501662343, 15.47447853739014068281273587723, 16.52703859754316162709641103359, 17.62587928152474285626729888015, 17.93500915747206319666017196151, 19.19321336464238330725203019207, 20.09895929283561986590686711635, 21.19669256012060190257643213244, 22.021237997376541611552782847090, 23.14750073950129190760497359372, 23.38991978111214323119594005355, 24.10248319136356016675173542084

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