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Properties

Label 1-671-671.151-r0-0-0
Degree $1$
Conductor $671$
Sign $-0.665 - 0.746i$
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  + (0.743 − 0.669i)2-s + (0.809 − 0.587i)3-s + (0.104 − 0.994i)4-s + (0.978 + 0.207i)5-s + (0.207 − 0.978i)6-s + (−0.406 − 0.913i)7-s + (−0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + (0.866 − 0.5i)10-s + (−0.5 − 0.866i)12-s + (−0.669 − 0.743i)13-s + (−0.913 − 0.406i)14-s + (0.913 − 0.406i)15-s + (−0.978 − 0.207i)16-s + (0.743 + 0.669i)17-s + (−0.406 − 0.913i)18-s + ⋯
L(s)  = 1  + (0.743 − 0.669i)2-s + (0.809 − 0.587i)3-s + (0.104 − 0.994i)4-s + (0.978 + 0.207i)5-s + (0.207 − 0.978i)6-s + (−0.406 − 0.913i)7-s + (−0.587 − 0.809i)8-s + (0.309 − 0.951i)9-s + (0.866 − 0.5i)10-s + (−0.5 − 0.866i)12-s + (−0.669 − 0.743i)13-s + (−0.913 − 0.406i)14-s + (0.913 − 0.406i)15-s + (−0.978 − 0.207i)16-s + (0.743 + 0.669i)17-s + (−0.406 − 0.913i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.665 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $-0.665 - 0.746i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ -0.665 - 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.269378358 - 2.830789547i\)
\(L(\frac12)\) \(\approx\) \(1.269378358 - 2.830789547i\)
\(L(1)\) \(\approx\) \(1.583446329 - 1.463257748i\)
\(L(1)\) \(\approx\) \(1.583446329 - 1.463257748i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (0.743 - 0.669i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.978 + 0.207i)T \)
7 \( 1 + (-0.406 - 0.913i)T \)
13 \( 1 + (-0.669 - 0.743i)T \)
17 \( 1 + (0.743 + 0.669i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + iT \)
29 \( 1 + (-0.994 - 0.104i)T \)
31 \( 1 + (0.207 + 0.978i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (-0.866 - 0.5i)T \)
47 \( 1 + (0.913 + 0.406i)T \)
53 \( 1 + (0.951 + 0.309i)T \)
59 \( 1 + (-0.406 - 0.913i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (-0.743 - 0.669i)T \)
73 \( 1 + (0.104 - 0.994i)T \)
79 \( 1 + (0.743 - 0.669i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 - iT \)
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

−22.76768007211010571381739175728, −22.12269678765569975555824203302, −21.60122657704746290941326811448, −20.84362538413259433465569500345, −20.19335316187036272444471172584, −18.87682592562259466595058153615, −18.165619729740988576633668645256, −16.83549421740540964365202351494, −16.44934104254229031884505813154, −15.490785806381346596758823359944, −14.71790666795275166249646886502, −14.04859157365676007360476359441, −13.36436131479899247199490082983, −12.46893913924943630684595292648, −11.58471904130813127442636152849, −10.10308022690815957621742524048, −9.27414962612875214622647829942, −8.79165339122132322026449505571, −7.58721442723575770934786353185, −6.664376364105743264657051745175, −5.465994952053257241615464484857, −5.04087761242277994793787188101, −3.81592325504731223450935333959, −2.72021790908285636674577043931, −2.155654036076308150234806840429, 1.12792093321542074578603073028, 1.89134042504882292308424292377, 3.17180730555002729817285885585, 3.5105161262325983769996484105, 5.04645582435938066065843149459, 5.96522980628004530367344250233, 6.91393765543764975317512177833, 7.73786471105574312759671602803, 9.184249909453300422293216000374, 10.00557740377349205832372160048, 10.402361731836712763137504440909, 11.83634735406545745641921389926, 12.69103552492937298433708154910, 13.40295953468644263231362083632, 13.91034017455819543693491171315, 14.653105035301424881012631987117, 15.45085636746759359249806946442, 16.84389496305028921038832445751, 17.743056482710608779821023599765, 18.637748835288773936826421362384, 19.38452810601762095551608085401, 20.2049595184735006557965828509, 20.664881130852632490879142065736, 21.61754669220734278993577988627, 22.36484915920143633843280776480

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