↑

Properties

Label 1-671-671.196-r0-0-0
Degree $1$
Conductor $671$
Sign $0.999 + 0.0244i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.104 + 0.994i)2-s + (0.309 − 0.951i)3-s + (−0.978 − 0.207i)4-s + (0.913 + 0.406i)5-s + (0.913 + 0.406i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.104 + 0.994i)13-s + (0.669 + 0.743i)14-s + (0.669 − 0.743i)15-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (0.669 − 0.743i)18-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)2-s + (0.309 − 0.951i)3-s + (−0.978 − 0.207i)4-s + (0.913 + 0.406i)5-s + (0.913 + 0.406i)6-s + (0.669 − 0.743i)7-s + (0.309 − 0.951i)8-s + (−0.809 − 0.587i)9-s + (−0.5 + 0.866i)10-s + (−0.5 + 0.866i)12-s + (−0.104 + 0.994i)13-s + (0.669 + 0.743i)14-s + (0.669 − 0.743i)15-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (0.669 − 0.743i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0244i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0244i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.999 + 0.0244i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ 0.999 + 0.0244i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.703529095 + 0.02078880413i\)
\(L(\frac12)\) \(\approx\) \(1.703529095 + 0.02078880413i\)
\(L(1)\) \(\approx\) \(1.258865440 + 0.1380030615i\)
\(L(1)\) \(\approx\) \(1.258865440 + 0.1380030615i\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−22.25585797102111243737292739303, −21.72561407501257019931153554472, −21.18144808049479420730837081110, −20.45612389413948464671549343446, −19.81586627171274705472697236013, −18.77669461732416219055328209470, −17.65168662785205736743271032473, −17.413836137025185505576191604347, −16.25527713690532505345006364131, −15.03978266508924040259153310317, −14.54512850508559677782791138485, −13.41809451015920334109082661067, −12.82813376310673449846659811032, −11.68264410982398346338160022639, −10.837357328760420545743665633, −10.15163770407974511729472000883, −9.2267128642394268589136861771, −8.745558491384083763600517072369, −7.84871034916761942960963277327, −5.80594304402629700932298204911, −5.1835990025241164408206597227, −4.427115853157499870635110221455, −3.10376556805130878257662739009, −2.420594351542733534062933444852, −1.26336559607335454838335832730, 1.00505117564879492931055724650, 2.00029650571345046777426737339, 3.388156393335637919719503600511, 4.725793861349219786963722654767, 5.67372990557685721113705742178, 6.67228918016474604893621556941, 7.20538268852585422032668933051, 7.99184896131874495516244601549, 9.0996456616893433612472460944, 9.72964819232053578296231726798, 10.96904509777626769129776426538, 12.018936168841489924341591784445, 13.32200312627269578738390362973, 13.74688461422694750018363724120, 14.32898455123733402880442625256, 15.04528437708191163601064066859, 16.43980619211220409390579541356, 17.194062468584243003195908890051, 17.690928155723150952332946287270, 18.697054670795374960782982437619, 18.93047774449991299662466906752, 20.41922937840169479716986537835, 21.03148329026997404057536187587, 22.333616005688246028463614596343, 22.907460586074847026796022247704

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