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Properties

Label 1-407-407.213-r0-0-0
Degree $1$
Conductor $407$
Sign $-0.406 - 0.913i$
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  + (0.615 + 0.788i)2-s + (0.559 + 0.829i)3-s + (−0.241 + 0.970i)4-s + (−0.990 − 0.139i)5-s + (−0.309 + 0.951i)6-s + (−0.719 − 0.694i)7-s + (−0.913 + 0.406i)8-s + (−0.374 + 0.927i)9-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.848 − 0.529i)13-s + (0.104 − 0.994i)14-s + (−0.438 − 0.898i)15-s + (−0.882 − 0.469i)16-s + (−0.848 + 0.529i)17-s + (−0.961 + 0.275i)18-s + ⋯
L(s)  = 1  + (0.615 + 0.788i)2-s + (0.559 + 0.829i)3-s + (−0.241 + 0.970i)4-s + (−0.990 − 0.139i)5-s + (−0.309 + 0.951i)6-s + (−0.719 − 0.694i)7-s + (−0.913 + 0.406i)8-s + (−0.374 + 0.927i)9-s + (−0.5 − 0.866i)10-s + (−0.939 + 0.342i)12-s + (−0.848 − 0.529i)13-s + (0.104 − 0.994i)14-s + (−0.438 − 0.898i)15-s + (−0.882 − 0.469i)16-s + (−0.848 + 0.529i)17-s + (−0.961 + 0.275i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 407 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.406 - 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(407\)    =    \(11 \cdot 37\)
Sign: $-0.406 - 0.913i$
Analytic conductor: \(1.89010\)
Root analytic conductor: \(1.89010\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{407} (213, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 407,\ (0:\ ),\ -0.406 - 0.913i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2638716322 + 0.4062690976i\)
\(L(\frac12)\) \(\approx\) \(-0.2638716322 + 0.4062690976i\)
\(L(1)\) \(\approx\) \(0.6448253302 + 0.6549801879i\)
\(L(1)\) \(\approx\) \(0.6448253302 + 0.6549801879i\)

Euler product

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

−23.55536275145669772989237006677, −22.912128235143431460506897555455, −22.11196412770431340241898878702, −21.04096519729634447373244671541, −20.00844886144690262861197525582, −19.50175451427128024363201395559, −18.79272860982460176092150442873, −18.22395229540683395364989249976, −16.61570904138204141365668202181, −15.32105487135606608726752797620, −14.87146678618481825810934366649, −13.82872191663259557778740967315, −12.87722864275506788706662058321, −12.13605401219707641754320983090, −11.67658030778947020895292520157, −10.338533757703829691718477996636, −9.18098844792644266007078330218, −8.42952029227174956759486755305, −7.01055540587253890043002903126, −6.35672769055870281001405014388, −4.89599992302928384162460382836, −3.74398183213334431017543784110, −2.82631019364499136771218321189, −1.98478693049188301772564862272, −0.19845145263400074153600145446, 2.75483091020948323466539566175, 3.62462039992319123868847326652, 4.385341467558928974042221049947, 5.2380053237637259687612523033, 6.76646573805096898096167637328, 7.534880989527603613052310742627, 8.48061677694029166082084668933, 9.346121301392649934399046219125, 10.57755036373525516471826714576, 11.5727469074250105982970050733, 12.89049249038819248183706962318, 13.38750422576021464906139704424, 14.70547892019772674644504792215, 15.2068853311910662769704278267, 15.94407921926394370114506488226, 16.73973962767086817030015263350, 17.475644136960039133636922726575, 19.14597471166169296224147008829, 19.9053322456701724262802808449, 20.507122511795291538088546548422, 21.84368273658115374902096837967, 22.24525751294733132800543141620, 23.25995043944227616155034175256, 23.91664289929233208687690191037, 24.93018037193881628933486949941

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