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Properties

Label 1-671-671.400-r0-0-0
Degree $1$
Conductor $671$
Sign $-0.993 - 0.111i$
Analytic cond. $3.11611$
Root an. cond. $3.11611$
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  + 2-s + (−0.809 + 0.587i)3-s + 4-s + (−0.809 + 0.587i)5-s + (−0.809 + 0.587i)6-s + (−0.809 − 0.587i)7-s + 8-s + (0.309 − 0.951i)9-s + (−0.809 + 0.587i)10-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)15-s + 16-s + (−0.809 + 0.587i)17-s + (0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + 2-s + (−0.809 + 0.587i)3-s + 4-s + (−0.809 + 0.587i)5-s + (−0.809 + 0.587i)6-s + (−0.809 − 0.587i)7-s + 8-s + (0.309 − 0.951i)9-s + (−0.809 + 0.587i)10-s + (−0.809 + 0.587i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 − 0.951i)15-s + 16-s + (−0.809 + 0.587i)17-s + (0.309 − 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $-0.993 - 0.111i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (400, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ -0.993 - 0.111i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02537087983 + 0.4552413534i\)
\(L(\frac12)\) \(\approx\) \(0.02537087983 + 0.4552413534i\)
\(L(1)\) \(\approx\) \(0.8938983092 + 0.3077995772i\)
\(L(1)\) \(\approx\) \(0.8938983092 + 0.3077995772i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
61 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (-0.809 + 0.587i)T \)
23 \( 1 + (-0.809 - 0.587i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (-0.809 - 0.587i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.809 - 0.587i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.809 + 0.587i)T \)
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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

−22.38775000196302241036716302878, −21.98473404013354072138432651679, −20.738609096849322573254186824, −19.878796191016204392003159003949, −19.29701504402566930630567688893, −18.32402040641135590525980130986, −17.20189185608392883663913120972, −16.38128568439484029692906290498, −15.63468319012312073088381373825, −15.20175339962866861380865066256, −13.6302129551096861135817189696, −12.934157162182528957726655143318, −12.542690940172118609245305399375, −11.53405467338034795754205345471, −11.10906390728781902047612396703, −9.82838795011538650466253499776, −8.42639027579069125294730561421, −7.52195049685683607756956865161, −6.616682353747701519946206545, −5.79751971179922110845907356641, −5.020490788802853341528264641303, −4.04211588195232274950354222207, −2.928665653697057753582730726525, −1.75486618288192848148949732297, −0.16218115982849275410723092070, 1.88031443702954971368226734556, 3.48060504671322851020087519472, 3.891147524427394034383920263037, 4.67927816173972267065852443206, 6.04318419281257829770793138342, 6.60480193030852046867217848504, 7.31396591532954020238430806994, 8.77209726489365560723971326291, 10.2062862690043014890440455521, 10.727231596204170667055620289737, 11.45783908605135967523718699837, 12.333589449666298455334394056388, 13.023271159320737521073484788781, 14.26257688635597707944397621431, 14.88470708460274397212539285489, 15.89882111682723213385060156963, 16.2756234660273404939486904150, 17.05394216373789259803002301104, 18.345449627120302744905180722243, 19.340339233349415661513563604626, 20.02688092175219726013975052582, 20.983017669568301110229213790448, 21.84952161924387826781656838636, 22.45579626640551587901510126580, 23.08011085715669048208402089575

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