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Properties

Label 1-61-61.48-r0-0-0
Degree $1$
Conductor $61$
Sign $0.462 + 0.886i$
Analytic cond. $0.283282$
Root an. cond. $0.283282$
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  + (0.5 + 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + 9-s + (0.5 − 0.866i)10-s − 11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + 3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (0.5 + 0.866i)6-s + (0.5 + 0.866i)7-s − 8-s + 9-s + (0.5 − 0.866i)10-s − 11-s + (−0.5 + 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.462 + 0.886i$
Analytic conductor: \(0.283282\)
Root analytic conductor: \(0.283282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (0:\ ),\ 0.462 + 0.886i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.131383620 + 0.6858246408i\)
\(L(\frac12)\) \(\approx\) \(1.131383620 + 0.6858246408i\)
\(L(1)\) \(\approx\) \(1.306892729 + 0.5799865204i\)
\(L(1)\) \(\approx\) \(1.306892729 + 0.5799865204i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−31.99188116392148569652143425592, −31.03876793177522121793180326945, −30.357175281733102095930080466147, −29.48129004384622402565696422515, −27.86479327911104326752742480727, −26.69577594244334007532406368691, −26.07302774612770157783581227490, −24.00939572750078286398961065942, −23.50537292645569377201249970082, −21.85807737601701341013502376276, −21.01330022864515481041508845981, −19.79056424652425608898750703856, −19.11245994943717360535158892025, −17.90693020071791309112458452600, −15.6553689182284219110761306530, −14.48838806802647004420570464831, −13.853029705849148073012538496834, −12.44359544768180114243884536944, −10.87191789262440677802401538174, −10.06992794730389083961293709993, −8.3172854906401107318267871354, −6.94285558532070839428565400323, −4.58625260794530546378074000939, −3.44392883425702581454096118525, −2.082633829785443739605837851793, 2.71090722267694431553152586772, 4.38539595882684609887470446619, 5.57182733280192471675925063704, 7.84331161013990297408994327117, 8.16035313439767258608874297113, 9.6099302065634368802028203685, 12.11566587734412236225392939208, 12.960337789352319690870473592775, 14.29205677048807924540351397186, 15.40798519804130329586156830211, 16.06934909591792193919298620008, 17.758054460919639425331223906906, 18.98174199547977574460303159874, 20.63722767772497200963889580612, 21.20773314214365924266523863264, 22.80608764120406999440779599212, 24.18324589175117322012554295721, 24.76391899438881796169598966447, 25.73026641846363910407617459322, 27.03626754340121736588527385870, 27.871520952546668105657475333768, 29.79926602129773772648480728535, 31.15841736469482571926864787082, 31.69199747752425351313728087761, 32.332397028582635743087533094439

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