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Properties

Label 1-1860-1860.1379-r1-0-0
Degree $1$
Conductor $1860$
Sign $-0.606 + 0.795i$
Analytic cond. $199.884$
Root an. cond. $199.884$
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  + (−0.809 − 0.587i)7-s + (0.809 + 0.587i)11-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.809 + 0.587i)23-s + (0.309 − 0.951i)29-s + 37-s + (−0.309 + 0.951i)41-s + (−0.309 + 0.951i)43-s + (−0.309 − 0.951i)47-s + (0.309 + 0.951i)49-s + (0.809 − 0.587i)53-s + (0.309 + 0.951i)59-s − 61-s + ⋯
L(s)  = 1  + (−0.809 − 0.587i)7-s + (0.809 + 0.587i)11-s + (0.309 + 0.951i)13-s + (0.809 − 0.587i)17-s + (−0.309 + 0.951i)19-s + (−0.809 + 0.587i)23-s + (0.309 − 0.951i)29-s + 37-s + (−0.309 + 0.951i)41-s + (−0.309 + 0.951i)43-s + (−0.309 − 0.951i)47-s + (0.309 + 0.951i)49-s + (0.809 − 0.587i)53-s + (0.309 + 0.951i)59-s − 61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.606 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.606 + 0.795i$
Analytic conductor: \(199.884\)
Root analytic conductor: \(199.884\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (1379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1860,\ (1:\ ),\ -0.606 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5001887698 + 1.010123988i\)
\(L(\frac12)\) \(\approx\) \(0.5001887698 + 1.010123988i\)
\(L(1)\) \(\approx\) \(0.9598643922 + 0.1120642647i\)
\(L(1)\) \(\approx\) \(0.9598643922 + 0.1120642647i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
31 \( 1 \)
good7 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (0.809 - 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + (-0.309 + 0.951i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 - T \)
67 \( 1 + T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.809 + 0.587i)T \)
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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

−19.71607210700107362279355331118, −18.93978956123947813857286006800, −18.33458009228165378923494353372, −17.44398271697707867985290900576, −16.703163376512873953521152888242, −15.99765251034194095731057963549, −15.32176546361066848370706815977, −14.54798939670555283317866593804, −13.758834551757468272488632966271, −12.85289842315913228792377064026, −12.37637046969592680424171374292, −11.522002048039487503094227881985, −10.60628017855710189948097475959, −9.967104291718593372116143447477, −8.966788533568821423527769905615, −8.52642198094854146889106641791, −7.51387110898457675392725178782, −6.4677970118788341523081663320, −5.989483637091612662726558873826, −5.143437540310503470067190051410, −3.939861923953435390431794650434, −3.25438736413827730043663945184, −2.431755349327496469310529750581, −1.17684087764602932744896769723, −0.22706043186951876098376018131, 1.03771528302774283634697671068, 1.88870241161024788761856556052, 3.090414236839992487779267056423, 3.95429315482062778275713234396, 4.48469736342900760563169527726, 5.829771829243756317155967104698, 6.4288651737135140159318002655, 7.21806088901027529443161427959, 7.98410829468415904911920168752, 9.0256085088071419578731160749, 9.91181756789895647573463124600, 10.06619461970510559385132482408, 11.52469659564498640328991205585, 11.84617462371047086836745443098, 12.82410500813660900114189427108, 13.58163432201412418947221006164, 14.26712263830687155088623167423, 14.92675207324766642362857932283, 16.00385093136602773611395013413, 16.56159205414957843980109870875, 17.050771112552938975125075765210, 18.09352873460503028561808211817, 18.75279789403821683428119904235, 19.593719943679585429771696770332, 20.01084276937293900010390498816

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