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Properties

Label 1-671-671.125-r0-0-0
Degree $1$
Conductor $671$
Sign $-0.954 - 0.298i$
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  + (−0.309 + 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 − 0.587i)4-s + 5-s − 6-s + (−0.309 + 0.951i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (0.309 − 0.951i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s − 17-s + (−0.309 − 0.951i)18-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)2-s + (0.309 + 0.951i)3-s + (−0.809 − 0.587i)4-s + 5-s − 6-s + (−0.309 + 0.951i)7-s + (0.809 − 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.309 + 0.951i)10-s + (0.309 − 0.951i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s − 17-s + (−0.309 − 0.951i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(671\)    =    \(11 \cdot 61\)
Sign: $-0.954 - 0.298i$
Analytic conductor: \(3.11611\)
Root analytic conductor: \(3.11611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{671} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 671,\ (0:\ ),\ -0.954 - 0.298i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1829332007 + 1.198104645i\)
\(L(\frac12)\) \(\approx\) \(-0.1829332007 + 1.198104645i\)
\(L(1)\) \(\approx\) \(0.5670084707 + 0.8317952469i\)
\(L(1)\) \(\approx\) \(0.5670084707 + 0.8317952469i\)

Euler product

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

−22.31276842360151910312536838946, −21.33000040610058530123846842700, −20.412086525536977957738528128777, −19.91614602085097215634124038870, −19.23548257059745159241906641283, −18.08215005925661546895941869426, −17.64305281156884695516361964477, −17.17728959083423938456727371524, −15.80799801043963515683543446072, −14.332288605953405999778830237429, −13.585991493275609592859286875761, −13.22891562077284345923357032089, −12.48049213115546817543175187905, −11.274347840391115578181070010609, −10.572477921655736141373340623063, −9.60156151987336329633324297755, −8.87858713462780244454825809464, −7.85440770047453452160444100852, −6.98369554779000641339693822110, −5.95522928451999729540028291483, −4.726367576842580694923408894762, −3.3886868866393061686838504218, −2.6187687609929277870653598010, −1.5858290884080093063572093997, −0.63871589453581509719767594561, 1.751034063266433276220225735175, 2.88306405157251535342628952712, 4.327505490727332842704528361938, 5.04609999517971368399916036773, 6.126562097262782737594707804527, 6.5236057650948248359761088660, 8.24680416890207116296720328803, 8.76243701082593188788257953121, 9.617716495754801350708321656115, 10.08036775425415494571557541457, 11.21313011224527695398732553928, 12.55561246027637259067359267986, 13.727389044411381146578933570186, 14.1927105036675494868545289985, 15.06608039792839967869458470490, 15.8958327663502955690695909275, 16.45060622361245313053455239055, 17.31458103012354872163359414235, 18.19536106300691647249587072237, 18.91195569970040583158143045900, 19.88691728382956577954659321088, 21.06053633310931425550913488948, 21.62734992495735419444439719401, 22.41422007935367797750529406123, 23.01936589978356319916234183470

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