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Properties

Label 1-61-61.38-r1-0-0
Degree $1$
Conductor $61$
Sign $0.128 - 0.991i$
Analytic cond. $6.55536$
Root an. cond. $6.55536$
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  + (−0.951 + 0.309i)2-s + (−0.309 − 0.951i)3-s + (0.809 − 0.587i)4-s + (0.809 − 0.587i)5-s + (0.587 + 0.809i)6-s + (0.951 + 0.309i)7-s + (−0.587 + 0.809i)8-s + (−0.809 + 0.587i)9-s + (−0.587 + 0.809i)10-s − i·11-s + (−0.809 − 0.587i)12-s + 13-s − 14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (0.587 + 0.809i)17-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (−0.309 − 0.951i)3-s + (0.809 − 0.587i)4-s + (0.809 − 0.587i)5-s + (0.587 + 0.809i)6-s + (0.951 + 0.309i)7-s + (−0.587 + 0.809i)8-s + (−0.809 + 0.587i)9-s + (−0.587 + 0.809i)10-s − i·11-s + (−0.809 − 0.587i)12-s + 13-s − 14-s + (−0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (0.587 + 0.809i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.128 - 0.991i$
Analytic conductor: \(6.55536\)
Root analytic conductor: \(6.55536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (38, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (1:\ ),\ 0.128 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8466480530 - 0.7437543014i\)
\(L(\frac12)\) \(\approx\) \(0.8466480530 - 0.7437543014i\)
\(L(1)\) \(\approx\) \(0.7864257517 - 0.3034351670i\)
\(L(1)\) \(\approx\) \(0.7864257517 - 0.3034351670i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 + (-0.951 + 0.309i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
11 \( 1 - iT \)
13 \( 1 + T \)
17 \( 1 + (0.587 + 0.809i)T \)
19 \( 1 + (-0.309 - 0.951i)T \)
23 \( 1 + (-0.587 - 0.809i)T \)
29 \( 1 - iT \)
31 \( 1 + (-0.951 - 0.309i)T \)
37 \( 1 + (-0.951 - 0.309i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + T \)
53 \( 1 + (0.587 - 0.809i)T \)
59 \( 1 + (0.951 - 0.309i)T \)
67 \( 1 + (0.587 + 0.809i)T \)
71 \( 1 + (-0.587 + 0.809i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (0.587 - 0.809i)T \)
83 \( 1 + (0.309 + 0.951i)T \)
89 \( 1 + (0.951 - 0.309i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−33.167474681948441125367859341308, −31.173300094538918697321314673291, −30.02834838859941485387573790242, −29.026109121576860421297482930084, −27.81475202901419104435498570215, −27.23063984692629299605267862604, −25.91759127299012251441584442316, −25.32547907454835941594892968712, −23.38938432594859230898765814311, −22.016612540525157506405415896079, −20.91781004668019655809372840631, −20.401501826294746587908902215290, −18.41607663486657084267770045620, −17.681197844767134126812190821654, −16.67312532499785719055444627751, −15.33287555036582994014520629891, −14.11494763320623576384522718010, −11.9999225673362675713475063099, −10.77251914684064633363992106386, −10.07935322930967539816651351432, −8.83905349540284493695281934035, −7.202143019437921568561556496174, −5.546696503761166676997840497398, −3.61355527197429726707790627429, −1.72643011936352240851948277252, 0.92114129091361188170945823031, 2.11694309824526000138562105110, 5.48680436971798510389747524073, 6.3416855378958896946909752582, 8.117651045018955270904927293986, 8.76888907563443764540819341970, 10.65307598452781362231188116892, 11.713339196837379849814762425381, 13.28209095798751050732962522608, 14.50753378403956427181604172391, 16.29284876923033455293092479520, 17.29345717412430153211070490256, 18.12116550942446222279179725506, 19.05632252241682839453404131368, 20.46795205027669932139205603227, 21.58919456044067829229228046604, 23.67009633521738204736615016658, 24.32162033697025696742474103925, 25.18788916450783126733107817311, 26.25687662870915789063121715559, 27.98103863143168196327873795360, 28.39862273542492431906611656321, 29.67388581454726936052257613520, 30.4536814079990765568218308133, 32.24698764988621002404706949173

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