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Properties

Label 1-671-671.153-r0-0-0
Degree $1$
Conductor $671$
Sign $-0.803 - 0.595i$
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  + (−0.994 + 0.104i)2-s + (0.809 − 0.587i)3-s + (0.978 − 0.207i)4-s + (−0.669 − 0.743i)5-s + (−0.743 + 0.669i)6-s + (−0.406 − 0.913i)7-s + (−0.951 + 0.309i)8-s + (0.309 − 0.951i)9-s + (0.743 + 0.669i)10-s + (0.669 − 0.743i)12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.978 − 0.207i)15-s + (0.913 − 0.406i)16-s + (−0.207 − 0.978i)17-s + (−0.207 + 0.978i)18-s + ⋯
L(s)  = 1  + (−0.994 + 0.104i)2-s + (0.809 − 0.587i)3-s + (0.978 − 0.207i)4-s + (−0.669 − 0.743i)5-s + (−0.743 + 0.669i)6-s + (−0.406 − 0.913i)7-s + (−0.951 + 0.309i)8-s + (0.309 − 0.951i)9-s + (0.743 + 0.669i)10-s + (0.669 − 0.743i)12-s + (0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.978 − 0.207i)15-s + (0.913 − 0.406i)16-s + (−0.207 − 0.978i)17-s + (−0.207 + 0.978i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $-0.803 - 0.595i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ -0.803 - 0.595i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3042730778 - 0.9218433353i\)
\(L(\frac12)\) \(\approx\) \(0.3042730778 - 0.9218433353i\)
\(L(1)\) \(\approx\) \(0.6865942725 - 0.4462143558i\)
\(L(1)\) \(\approx\) \(0.6865942725 - 0.4462143558i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
61 \( 1 \)
good2 \( 1 + (-0.994 + 0.104i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (-0.669 - 0.743i)T \)
7 \( 1 + (-0.406 - 0.913i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.207 - 0.978i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 + (0.951 + 0.309i)T \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (0.994 + 0.104i)T \)
37 \( 1 + (-0.587 + 0.809i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.207 - 0.978i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.951 + 0.309i)T \)
59 \( 1 + (-0.994 + 0.104i)T \)
67 \( 1 + (0.743 - 0.669i)T \)
71 \( 1 + (-0.743 - 0.669i)T \)
73 \( 1 + (-0.669 + 0.743i)T \)
79 \( 1 + (-0.207 + 0.978i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (0.587 + 0.809i)T \)
97 \( 1 + (0.104 - 0.994i)T \)
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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

−23.01225556392391980103237809054, −21.898113869380419114595583608250, −21.44595456603085556425639231822, −20.48567311397104229579122122461, −19.50726084348911925894576141056, −19.140382142099343463212113240196, −18.49129207573247923818949702423, −17.47614506652063632060044643698, −16.11544537924657787824565325352, −15.89092450487395073657837472534, −15.02344934468534232988595441664, −14.34606058112434100908354208695, −13.014883921244734457309912213414, −11.90714016537231107518884220717, −11.13675958018372127643071923528, −10.34339660191302414508361818480, −9.413332715982954338703915130972, −8.717067360913668023684888105325, −8.027103248532160902569577329342, −6.971423209445049541193484487557, −6.17725495782965857769743712941, −4.58410658672043072464784432766, −3.28522353534569489646775043549, −2.85164388149257648234173165531, −1.65673523274825806095112536290, 0.65108381994022461234303757882, 1.33401633821051263049688347469, 2.91398553004868624152913967520, 3.58904572172473103829282954910, 5.07660733189060619345849903807, 6.50199765347606910955292107598, 7.30464628321238668761100665062, 7.94933603082774667510966201959, 8.68562480636450235476376687210, 9.55095405450092399128158440139, 10.37811147030079495256690597827, 11.57999058320702356141360282551, 12.29730860074978380513370931668, 13.33244236675034199993881702519, 14.02219458295755677333094079694, 15.49503154365898307730999264103, 15.667331515473072606050293255411, 16.82031123893023441826979180814, 17.53595446401084834845577714022, 18.50924452863568290095340638847, 19.219384170112921403233777842838, 19.97657830955103401186313123598, 20.47015882963958803394449281642, 20.96112617589280096962650612175, 22.80849515276815100602132506149

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