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Properties

Label 1-61-61.5-r0-0-0
Degree $1$
Conductor $61$
Sign $-0.100 + 0.994i$
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  + (−0.669 + 0.743i)2-s + (0.309 + 0.951i)3-s + (−0.104 − 0.994i)4-s + (0.913 + 0.406i)5-s + (−0.913 − 0.406i)6-s + (0.978 − 0.207i)7-s + (0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.913 + 0.406i)10-s − 11-s + (0.913 − 0.406i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.104 + 0.994i)15-s + (−0.978 + 0.207i)16-s + (0.104 + 0.994i)17-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (0.309 + 0.951i)3-s + (−0.104 − 0.994i)4-s + (0.913 + 0.406i)5-s + (−0.913 − 0.406i)6-s + (0.978 − 0.207i)7-s + (0.809 + 0.587i)8-s + (−0.809 + 0.587i)9-s + (−0.913 + 0.406i)10-s − 11-s + (0.913 − 0.406i)12-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)14-s + (−0.104 + 0.994i)15-s + (−0.978 + 0.207i)16-s + (0.104 + 0.994i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(61\)
Sign: $-0.100 + 0.994i$
Analytic conductor: \(0.283282\)
Root analytic conductor: \(0.283282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{61} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 61,\ (0:\ ),\ -0.100 + 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5451995669 + 0.6029503673i\)
\(L(\frac12)\) \(\approx\) \(0.5451995669 + 0.6029503673i\)
\(L(1)\) \(\approx\) \(0.7482076253 + 0.5125493758i\)
\(L(1)\) \(\approx\) \(0.7482076253 + 0.5125493758i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad61 \( 1 \)
good2 \( 1 + (-0.669 + 0.743i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (0.913 + 0.406i)T \)
7 \( 1 + (0.978 - 0.207i)T \)
11 \( 1 - T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (-0.978 - 0.207i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (0.104 - 0.994i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
67 \( 1 + (-0.913 - 0.406i)T \)
71 \( 1 + (-0.913 + 0.406i)T \)
73 \( 1 + (0.913 - 0.406i)T \)
79 \( 1 + (0.104 - 0.994i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.669 + 0.743i)T \)
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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

−31.63032012285317097856245976216, −31.07967531864408093246346353549, −29.664593285799345352330127954462, −29.175386406127161954306775422712, −28.07048894504748056838273554363, −26.68351748260581963709635995783, −25.52101811005130630704624925282, −24.70356907184262223936724633660, −23.41274889412016046662951971737, −21.4681193701626490752760340177, −20.91173832213128542078382463970, −19.63855600849365428574093026289, −18.346803864854079230342968980749, −17.79753982268963355204983134855, −16.60553932829123426668508292799, −14.390613924944570226968774686, −13.25817048943825931911324749619, −12.24974991982701756554472967589, −10.96296922982038716909987574191, −9.35089890341811762034608095111, −8.35342973849612087331020206411, −7.07545766295571238266962713767, −5.04903541065735815784456018084, −2.59614927947992060354621861199, −1.56252436725873366610397198561, 2.326456695812353284848099762628, 4.76884256956956601295569032986, 5.84588125057169941271758671134, 7.72561211785368057714718512107, 8.82364257793633560234347341685, 10.346620337649597958005143151644, 10.68666107246087516877206298282, 13.45010175300192820063184750925, 14.73697617323359658269645799654, 15.26332479602001043775246470935, 16.955848642418354184799554112969, 17.57599841703519920694471159388, 18.94338796589199262894495814297, 20.46788586443802844127460981657, 21.38953428628502924084576305136, 22.74258942724319014940113049069, 24.13294992230035122106112836189, 25.37943137306110139816620996398, 26.15426183319436634450007248788, 27.10226935620048051728886110948, 28.03971999555484298383730794102, 29.21552982728841446530521064660, 30.75182386510041113868484189507, 32.24052556274513396968413255226, 32.97173788366851420674289753954

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