↑

Properties

Label 1-671-671.160-r0-0-0
Degree $1$
Conductor $671$
Sign $0.893 + 0.448i$
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  + (−0.587 + 0.809i)2-s + (0.809 − 0.587i)3-s + (−0.309 − 0.951i)4-s − 5-s + i·6-s + (−0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s + (0.309 − 0.951i)9-s + (0.587 − 0.809i)10-s + (−0.809 − 0.587i)12-s + (0.809 − 0.587i)13-s + (−0.309 − 0.951i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + i·17-s + (0.587 + 0.809i)18-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (0.809 − 0.587i)3-s + (−0.309 − 0.951i)4-s − 5-s + i·6-s + (−0.587 + 0.809i)7-s + (0.951 + 0.309i)8-s + (0.309 − 0.951i)9-s + (0.587 − 0.809i)10-s + (−0.809 − 0.587i)12-s + (0.809 − 0.587i)13-s + (−0.309 − 0.951i)14-s + (−0.809 + 0.587i)15-s + (−0.809 + 0.587i)16-s + i·17-s + (0.587 + 0.809i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $0.893 + 0.448i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ 0.893 + 0.448i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9924939926 + 0.2351924358i\)
\(L(\frac12)\) \(\approx\) \(0.9924939926 + 0.2351924358i\)
\(L(1)\) \(\approx\) \(0.8368537627 + 0.1530404045i\)
\(L(1)\) \(\approx\) \(0.8368537627 + 0.1530404045i\)

Euler product

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

−22.74888126494339716276437294451, −21.50111979329384212013013371635, −20.96831697579110956124228617894, −20.04566033985246786221601547523, −19.550969556056270832602275827254, −19.06658705418870348124406252907, −17.9870222735689070737649745664, −16.8494732255417643229146163955, −15.99536643778180688835760778109, −15.68156566461005103935413700242, −14.1546545589346657344106897362, −13.577924796082839888390888600507, −12.635559991586300149911624731666, −11.534339156012409698292980943040, −10.86370095445948138188541477292, −10.01834476133336791137952950377, −9.17236891520896863854808622343, −8.41659695714238696797165969930, −7.582918054899083192851306860094, −6.76700163973751427603207949786, −4.68238728570883890999574767961, −4.00197037128770750712903578407, −3.3443001413947849605729888071, −2.35416784941197177922608109771, −0.82085240480993505666285032598, 0.86100773630947575066167533530, 2.22332144129979189662565242344, 3.435226874986335692462187014301, 4.43734661140189532071050692390, 6.04535258629485783081031219030, 6.469901959196528450966710747810, 7.65614426675653442378328575801, 8.45990029113468454530767431246, 8.65689891284319021307613188809, 9.947050483023424967028651774867, 10.84444052091577146945458396558, 12.24815690377522187807534240120, 12.76734781305236722994818043032, 13.921936291485361885347880265813, 14.779673452676166824442966395457, 15.51771424542000341539032181481, 15.903504505346491843573063454446, 17.10285781151631656860466953918, 18.11545971774286321324835753034, 18.8900086268570805572725676470, 19.22189711113097869868545326024, 20.04026592982406943271158406819, 20.92467448367804585422476530809, 22.32043855387533995492754851435, 23.15062845595924936880495412486

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