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Properties

Label 1-1800-1800.59-r0-0-0
Degree $1$
Conductor $1800$
Sign $0.704 - 0.709i$
Analytic cond. $8.35916$
Root an. cond. $8.35916$
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  + (−0.5 + 0.866i)7-s + (0.978 + 0.207i)11-s + (−0.978 + 0.207i)13-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.669 + 0.743i)23-s + (−0.104 − 0.994i)29-s + (0.104 − 0.994i)31-s + (0.309 − 0.951i)37-s + (0.978 − 0.207i)41-s + (0.5 − 0.866i)43-s + (0.104 + 0.994i)47-s + (−0.5 − 0.866i)49-s + (0.809 − 0.587i)53-s + (0.978 − 0.207i)59-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)7-s + (0.978 + 0.207i)11-s + (−0.978 + 0.207i)13-s + (−0.809 − 0.587i)17-s + (−0.809 − 0.587i)19-s + (−0.669 + 0.743i)23-s + (−0.104 − 0.994i)29-s + (0.104 − 0.994i)31-s + (0.309 − 0.951i)37-s + (0.978 − 0.207i)41-s + (0.5 − 0.866i)43-s + (0.104 + 0.994i)47-s + (−0.5 − 0.866i)49-s + (0.809 − 0.587i)53-s + (0.978 − 0.207i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.704 - 0.709i$
Analytic conductor: \(8.35916\)
Root analytic conductor: \(8.35916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (0:\ ),\ 0.704 - 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.027103232 - 0.4275418108i\)
\(L(\frac12)\) \(\approx\) \(1.027103232 - 0.4275418108i\)
\(L(1)\) \(\approx\) \(0.9291205232 + 0.01275877398i\)
\(L(1)\) \(\approx\) \(0.9291205232 + 0.01275877398i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.978 + 0.207i)T \)
13 \( 1 + (-0.978 + 0.207i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.669 + 0.743i)T \)
29 \( 1 + (-0.104 - 0.994i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (0.978 - 0.207i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (0.104 + 0.994i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.978 - 0.207i)T \)
61 \( 1 + (0.978 + 0.207i)T \)
67 \( 1 + (0.104 - 0.994i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (0.104 + 0.994i)T \)
83 \( 1 + (0.913 - 0.406i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.104 + 0.994i)T \)
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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

−20.03853671122189671145771296540, −19.670688440562386609363768207911, −19.04052232071214029554931070513, −17.92731292120082453795463679784, −17.298740090560240027174196145279, −16.60734543828966327967215219700, −16.100476344505751558332479800060, −14.8516932807257989536336305264, −14.510324392796024776313579682884, −13.61816559209301317164090071783, −12.77119056947466925285071462133, −12.23716396424071641136770471815, −11.22875001394572572167303917589, −10.41804093712964388965940813103, −9.89559747430292640417078500563, −8.894763985969988261355380670560, −8.211262411375715411819305989737, −7.11680407711391294733819429750, −6.62039012662900805406745888176, −5.79231724724445876002881566850, −4.523138183247230619116042519941, −4.0443618780320481671612202078, −3.061793687184466648181343508199, −2.006246052062899966683307822507, −0.923563628606952795530566917817, 0.46775391537895691160162973597, 2.16024702919748377134767484424, 2.430905069690093568328756010978, 3.814668388051827791625945165543, 4.47128051921847023077438621212, 5.53346582810865885078857534307, 6.2893319164555092307168450603, 7.030207963806101496864589238223, 7.88790096353740207573072698115, 9.1233719992353042320595914145, 9.260418739375843233769694739555, 10.18054115004643135334162245186, 11.384140743432060523023475862949, 11.797456664299385171419709085735, 12.63413157501773734330200706706, 13.33310153695511774134292067936, 14.28133591335174186887982292930, 14.972618415805690838209311120669, 15.634983138478694403978731367608, 16.35917890821525880774177028709, 17.36394714080026982658609878088, 17.6889158702951476348453124455, 18.80466309134208535833637181677, 19.44174573362615587830329305945, 19.83835293975147858064314457460

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