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Properties

Label 1-4620-4620.4199-r0-0-0
Degree $1$
Conductor $4620$
Sign $0.530 + 0.847i$
Analytic cond. $21.4551$
Root an. cond. $21.4551$
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  + (−0.309 + 0.951i)13-s + (0.309 + 0.951i)17-s + (0.809 − 0.587i)19-s − 23-s + (−0.809 − 0.587i)29-s + (0.309 − 0.951i)31-s + (0.809 + 0.587i)37-s + (0.809 − 0.587i)41-s − 43-s + (−0.809 + 0.587i)47-s + (0.309 − 0.951i)53-s + (0.809 + 0.587i)59-s + (0.309 + 0.951i)61-s + 67-s + (0.309 + 0.951i)71-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)13-s + (0.309 + 0.951i)17-s + (0.809 − 0.587i)19-s − 23-s + (−0.809 − 0.587i)29-s + (0.309 − 0.951i)31-s + (0.809 + 0.587i)37-s + (0.809 − 0.587i)41-s − 43-s + (−0.809 + 0.587i)47-s + (0.309 − 0.951i)53-s + (0.809 + 0.587i)59-s + (0.309 + 0.951i)61-s + 67-s + (0.309 + 0.951i)71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4620 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.530 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4620\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.530 + 0.847i$
Analytic conductor: \(21.4551\)
Root analytic conductor: \(21.4551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4620} (4199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4620,\ (0:\ ),\ 0.530 + 0.847i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.330120133 + 0.7370545386i\)
\(L(\frac12)\) \(\approx\) \(1.330120133 + 0.7370545386i\)
\(L(1)\) \(\approx\) \(1.037769759 + 0.1187993680i\)
\(L(1)\) \(\approx\) \(1.037769759 + 0.1187993680i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (0.809 - 0.587i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.809 - 0.587i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.309 - 0.951i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + T \)
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

−18.213091944145914178840301971958, −17.548120504010041933938809868883, −16.59025355741376306917593885340, −16.183787397619133360834600919693, −15.46110894276986292022375833112, −14.664174010984691788713244139601, −14.126382669936519472053365864492, −13.41876554261690717399392760794, −12.60325537844704735947896263696, −12.095934624439153538333370337655, −11.325055384434047838156475657979, −10.61559248676993526690764024503, −9.7578960969351038234243373522, −9.46646683795736824416109849716, −8.29286394097071280883294852449, −7.84055676395319180980363545427, −7.10950047597443029383293231054, −6.28061151873557605489228450554, −5.379379614868569236126824835048, −5.02721056313676549591574676620, −3.87619054395479145838207989611, −3.23446296630648414819214035859, −2.45245494544933138878162286551, −1.47286221060977444551983827229, −0.50244290028224531963603797690, 0.865665146178972338090263285246, 1.89713267463997739787962056302, 2.51082342823736631163749915651, 3.63279388205825080402139775118, 4.16544044943962507895736607870, 5.023087161530158158529117142, 5.86263591179882425575643856269, 6.47607243439908869522290271186, 7.32401040467973488141680333833, 7.977580849554990426192062074425, 8.68683508417931779153832958665, 9.66824362629053397174991300222, 9.8732567309055286937931835059, 10.9505802582860964913672567687, 11.63770824116365250435527338461, 12.04357209282766289975831105310, 13.12833293316016098976793590135, 13.4352115670303420981457860375, 14.49001311505241542835481413668, 14.756022915291081378189665254119, 15.74443065744040283022700463789, 16.27135218285770741599199531962, 17.03337870551477363131208636331, 17.53133249705721287529162128815, 18.41337025241157184162942479423

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