Properties

Label 1-60-60.23-r1-0-0
Degree $1$
Conductor $60$
Sign $0.525 + 0.850i$
Analytic cond. $6.44789$
Root an. cond. $6.44789$
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  + i·7-s + 11-s + i·13-s + i·17-s + 19-s + i·23-s + 29-s − 31-s i·37-s − 41-s i·43-s i·47-s − 49-s i·53-s − 59-s + ⋯
L(s)  = 1  + i·7-s + 11-s + i·13-s + i·17-s + 19-s + i·23-s + 29-s − 31-s i·37-s − 41-s i·43-s i·47-s − 49-s i·53-s − 59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(60\)    =    \(2^{2} \cdot 3 \cdot 5\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(6.44789\)
Root analytic conductor: \(6.44789\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{60} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 60,\ (1:\ ),\ 0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.363271944 + 0.7600738898i\)
\(L(\frac12)\) \(\approx\) \(1.363271944 + 0.7600738898i\)
\(L(1)\) \(\approx\) \(1.116460088 + 0.2635604750i\)
\(L(1)\) \(\approx\) \(1.116460088 + 0.2635604750i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 + iT \)
23 \( 1 \)
29 \( 1 \)
31 \( 1 \)
37 \( 1 + T \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 + iT \)
67 \( 1 \)
71 \( 1 + T \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 \)
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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

−32.47629831230629100812421841896, −30.907939058767231460173794719320, −29.97382155941892729175927550649, −29.01199099200447375228432096691, −27.49516862742607678282041252595, −26.80061997153399857770338446715, −25.37943164563552055156515775039, −24.3819461446618549027204789329, −23.018041811990797061857960197242, −22.20392889426890748651969139337, −20.509898229457938191204164318, −19.88179889932927834925671167357, −18.303996394996416542290853958576, −17.15279168710741169213021042506, −16.060828310768384395484329900880, −14.528422855887375320598507137581, −13.51484174190843534568873420028, −12.056604039931081751720940057405, −10.70600172976068715971985695904, −9.46342905761415244252004566005, −7.83450379851490714324816393489, −6.574931414236704717195471196598, −4.79398535490606614833976040171, −3.24253040212361290932886942932, −0.93760066860586079967577664072, 1.79968482505035636152863134721, 3.71653708964737002086467968890, 5.47816587925655604170977029377, 6.8425013612784414425580565662, 8.589462436093642181770670963290, 9.61791999663501914840167904495, 11.40754165149469174925118454004, 12.32069625266371446806383647334, 13.91951671939523924208441791961, 15.06444370483554258066095814268, 16.31371676524342624117768262889, 17.60260492994251618537524405156, 18.85906003450786626718290381874, 19.846235195192252619579427882340, 21.45339972046547542390903153499, 22.123496157108148960640535132452, 23.62328990345090387457648273806, 24.736747703131397346404269467684, 25.719698993245190667001009162838, 27.03753658852607193933408954791, 28.15841989838870233407690190442, 29.06648280965553959512787694656, 30.49060958521061426082003844854, 31.346239728689836989941297184092, 32.53832131391539133309664273823

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