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Properties

Label 1-1800-1800.29-r1-0-0
Degree $1$
Conductor $1800$
Sign $0.959 + 0.282i$
Analytic cond. $193.436$
Root an. cond. $193.436$
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 + (−0.104 + 0.994i)11-s + (−0.104 − 0.994i)13-s + (0.309 − 0.951i)17-s + (−0.309 + 0.951i)19-s + (0.913 − 0.406i)23-s + (0.669 − 0.743i)29-s + (0.669 + 0.743i)31-s + (−0.809 + 0.587i)37-s + (0.104 + 0.994i)41-s + (−0.5 − 0.866i)43-s + (0.669 − 0.743i)47-s + (−0.5 + 0.866i)49-s + (−0.309 − 0.951i)53-s + (−0.104 − 0.994i)59-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)7-s + (−0.104 + 0.994i)11-s + (−0.104 − 0.994i)13-s + (0.309 − 0.951i)17-s + (−0.309 + 0.951i)19-s + (0.913 − 0.406i)23-s + (0.669 − 0.743i)29-s + (0.669 + 0.743i)31-s + (−0.809 + 0.587i)37-s + (0.104 + 0.994i)41-s + (−0.5 − 0.866i)43-s + (0.669 − 0.743i)47-s + (−0.5 + 0.866i)49-s + (−0.309 − 0.951i)53-s + (−0.104 − 0.994i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.959 + 0.282i$
Analytic conductor: \(193.436\)
Root analytic conductor: \(193.436\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (1:\ ),\ 0.959 + 0.282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.416591617 + 0.3482361718i\)
\(L(\frac12)\) \(\approx\) \(2.416591617 + 0.3482361718i\)
\(L(1)\) \(\approx\) \(1.189382143 + 0.1074782775i\)
\(L(1)\) \(\approx\) \(1.189382143 + 0.1074782775i\)

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 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (-0.104 - 0.994i)T \)
17 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.913 - 0.406i)T \)
29 \( 1 + (0.669 - 0.743i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.104 + 0.994i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.669 - 0.743i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (0.104 - 0.994i)T \)
67 \( 1 + (0.669 + 0.743i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (0.978 - 0.207i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
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

−19.77626754161007905838230218926, −19.33914295721727072752157902681, −18.63659273787789757954099568792, −17.57301992664724054780920487941, −17.07417288804164073850131606819, −16.438178335358231843142973507737, −15.560438935571711077746371610541, −14.71826469423927266418633612273, −13.94022421813653585459506471142, −13.49317171999922552895009991374, −12.55769732449161981149239825510, −11.60668198212665240009224744900, −10.913405456075026435923030424679, −10.44089489262370276573885657086, −9.25977144330119908493847447784, −8.64688590447090382746194232670, −7.77251763649851584875694010548, −6.97549443037652826180003963084, −6.233005222034796741426160404209, −5.20751467061199290597985268432, −4.377818106049017743112046031344, −3.63531682571809345368145318135, −2.618711211801307604146819289809, −1.466494364222544403252254178662, −0.68210202090188151797952199085, 0.656935413693195146615128038754, 1.81390245986476943025912801912, 2.62660408474556874566579369550, 3.487705752908147318970248965264, 4.89543118901964354282942019164, 5.07119744162492904700611359835, 6.209867082808206267093226305768, 7.05719770217663377105124329782, 8.02255276514332356129257011086, 8.50250867976035483880294602715, 9.59800088395750066226910367442, 10.16271438049353608372813965133, 11.053115019468984325678216769234, 12.12844739361820318811814369751, 12.29822443350779263544728823727, 13.322837834840522595736727604993, 14.23380282250304014211431042760, 15.00861647277881327671976473787, 15.4357908994466017479091593719, 16.29309425329255502020416017065, 17.332452895843680847865706067009, 17.77674537168930012194925977833, 18.60657880766788174670800651219, 19.131382098312471518204732117806, 20.29622252525349892693875111648

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