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Properties

Label 1-1800-1800.1397-r0-0-0
Degree $1$
Conductor $1800$
Sign $0.959 + 0.282i$
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  + (0.866 − 0.5i)7-s + (−0.104 + 0.994i)11-s + (0.994 − 0.104i)13-s + (−0.951 − 0.309i)17-s + (0.309 − 0.951i)19-s + (0.406 + 0.913i)23-s + (−0.669 + 0.743i)29-s + (0.669 + 0.743i)31-s + (0.587 + 0.809i)37-s + (0.104 + 0.994i)41-s + (0.866 − 0.5i)43-s + (−0.743 − 0.669i)47-s + (0.5 − 0.866i)49-s + (0.951 − 0.309i)53-s + (0.104 + 0.994i)59-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)7-s + (−0.104 + 0.994i)11-s + (0.994 − 0.104i)13-s + (−0.951 − 0.309i)17-s + (0.309 − 0.951i)19-s + (0.406 + 0.913i)23-s + (−0.669 + 0.743i)29-s + (0.669 + 0.743i)31-s + (0.587 + 0.809i)37-s + (0.104 + 0.994i)41-s + (0.866 − 0.5i)43-s + (−0.743 − 0.669i)47-s + (0.5 − 0.866i)49-s + (0.951 − 0.309i)53-s + (0.104 + 0.994i)59-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(1800\)    =    \(2^{3} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.959 + 0.282i$
Analytic conductor: \(8.35916\)
Root analytic conductor: \(8.35916\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1800} (1397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1800,\ (0:\ ),\ 0.959 + 0.282i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.847983913 + 0.2662985500i\)
\(L(\frac12)\) \(\approx\) \(1.847983913 + 0.2662985500i\)
\(L(1)\) \(\approx\) \(1.235565828 + 0.04246657786i\)
\(L(1)\) \(\approx\) \(1.235565828 + 0.04246657786i\)

Euler product

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

−20.42223448743620283706521699262, −19.15181324739482124141126844486, −18.776312416520039316161110078942, −17.99547602559061703298816814562, −17.31559993783116826707627775461, −16.37463309750834150699442246555, −15.81262533717976117738215997746, −14.93376310206875345969847569120, −14.27628823581241663728118427117, −13.49590043629245910274507512082, −12.795816234224110720140079099127, −11.774550590391012300187911823578, −11.15739259663840888687170682101, −10.64586852574220787160138303294, −9.47955890322359693460835037165, −8.58073285224250789236762172352, −8.2567613641270454846240506069, −7.25616555092190026352635433676, −5.93646505810434610015794423428, −5.86864411941654951713365396056, −4.52811528942943480010711911349, −3.87907191438302998582746289300, −2.74294457443308528420596534191, −1.88681700012537399087461022727, −0.8171921825598336629619091049, 1.01765400369113447449820536411, 1.83670332311418505353454720918, 2.89856770835127111446991662894, 3.96825091615868697179016713807, 4.74158351863034149694055686925, 5.3739094611250708609673974828, 6.64864500857914433700349316440, 7.182281947887956025277505709689, 8.05927036665803146506525726589, 8.859543230758386777919133154441, 9.63926953276175801991238942433, 10.607128221624139336058978290059, 11.23787922442598969784493563863, 11.8213450979820485438042700344, 13.02766864598153938574645287159, 13.44594738562847646147226452485, 14.28509336638328783022893045634, 15.194156083066856757265323045539, 15.61676835474151599544253203723, 16.61130463607794436050313787996, 17.49749125725854018408095049976, 17.92473637399800507750619382853, 18.54002945867696118595433305239, 19.889666773129460255179097560832, 20.03540468527943450742686311419

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