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Properties

Label 1-1800-1800.491-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: $\chi_{1800} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (0:\ ),\ 0.704 - 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.836444130 - 0.7644379108i\)
\(L(\frac12)\) \(\approx\) \(1.836444130 - 0.7644379108i\)
\(L(1)\) \(\approx\) \(1.272027344 - 0.2085380488i\)
\(L(1)\) \(\approx\) \(1.272027344 - 0.2085380488i\)

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

−20.43241028290452237879346105574, −19.24244235608165722099761172856, −19.01851417714444626032108041787, −18.01126350868899183492676472990, −17.515580154659995749577710351357, −16.43581817586108689541345854795, −16.02582316092615417613916377446, −14.96445084345423569854012185246, −14.437181850653371062835584088233, −13.764251801232205693606757733763, −12.67123605374500753132570197859, −12.10452804014166970796582918826, −11.29149237070561389146300619518, −10.72624806572051749529469406177, −9.52907503639507632969392636161, −8.91326363069274739374285756711, −8.3118828490777754784166645019, −7.30416907327795492686785078315, −6.39871208163125658902440916077, −5.66642951096124471114309903436, −4.89211741987964828106842744489, −3.79732523075564608288923005315, −3.11294503383069854646071308256, −1.86025524553836484842294170261, −1.18409231520198366576963492895, 0.83696403609535956967696598495, 1.58469877530715730775249538149, 2.78536591033889801555997014589, 3.98987449709396518288546719786, 4.25456200966008976829494434514, 5.46921134329867352424592800090, 6.40262404603931183669897994967, 7.00201887235656659976918857989, 8.08224489174388195743978583045, 8.55271180473757692187367309190, 9.632390387651781872353494916009, 10.35387011385857293921335087032, 11.17593272227560171646066576574, 11.68489864330101083810177356143, 12.84628174578795787309069931845, 13.333622609514470924437913325090, 14.30408663415389804628430808085, 14.793181812801880056279963549, 15.614516977876654888110879106794, 16.70073488664387572980631292924, 17.07900891452196608320764541166, 17.742139335282963892510260669173, 18.76298657688193609419839332450, 19.32090714302923659960431032670, 20.199752502296717505726347606000

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