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Properties

Label 1-61-61.21-r1-0-0
Degree $1$
Conductor $61$
Sign $0.404 - 0.914i$
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  + (0.866 − 0.5i)2-s − 3-s + (0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.866 + 0.5i)6-s + (0.866 − 0.5i)7-s − i·8-s + 9-s + (0.866 + 0.5i)10-s − i·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 + (0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s − 3-s + (0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.866 + 0.5i)6-s + (0.866 − 0.5i)7-s − i·8-s + 9-s + (0.866 + 0.5i)10-s − i·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 + (0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.404 - 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $0.404 - 0.914i$
Analytic conductor: \(6.55536\)
Root analytic conductor: \(6.55536\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (1:\ ),\ 0.404 - 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.869609202 - 1.217069819i\)
\(L(\frac12)\) \(\approx\) \(1.869609202 - 1.217069819i\)
\(L(1)\) \(\approx\) \(1.449678871 - 0.5601487158i\)
\(L(1)\) \(\approx\) \(1.449678871 - 0.5601487158i\)

Euler product

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

−32.64729090447486601154946361052, −31.47610788868744712090366749225, −30.37622625547569420325551651299, −29.12514923060750577839100374090, −28.271964840298968033875486464178, −26.982192110726000934300097090733, −25.22063924895309818716141427065, −24.485409894684768320299298242677, −23.54153046877715196043286011238, −22.42632930954107561993079099650, −21.29515567521427893524888588167, −20.61989987517099058327420829247, −18.30016034726480525828214252140, −17.16640756475607871184591217203, −16.45025002864547861755641228380, −15.08472058401320899972311746229, −13.77220740299176146943860381632, −12.19280589375928219526952233766, −11.92642167202349199381764642902, −9.9066445986415140374528278483, −8.06479533540412486120966784512, −6.54952555148508768719433450059, −5.18533412323191564927241177579, −4.57863895810654845465502592596, −1.841566963539906552304640576278, 1.20896930369786053844077686797, 3.21106011319452850794718361931, 4.989505130968600195908500917301, 5.98973915763660261603061043168, 7.35649826668744702075018947071, 10.102047734233666192445767455280, 10.87483593342609957452230287692, 11.81892280330322887394175684123, 13.329794993522705623603335975338, 14.36826440684226754177383712492, 15.63487122521315153274563766928, 17.23551502609732742616343769070, 18.27491515494216684189221407643, 19.59370410514749467220076179674, 21.304380030079776509407353376721, 21.79272681367932077547395242547, 22.977598572797908934978208221570, 23.836260162495216051629521370932, 24.929716157981211555451339033562, 26.81059040682833073877657154273, 27.80519822048092982593714882025, 29.14319717245726529359063136076, 29.92975184128808482438193150538, 30.49277776945965643433756959869, 32.23547215903778510006111038738

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