Properties

Label 1-2205-2205.194-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.371 + 0.928i$
Analytic cond. $10.2399$
Root an. cond. $10.2399$
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  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)8-s + (−0.826 − 0.563i)11-s + (0.826 + 0.563i)13-s + (−0.222 − 0.974i)16-s + (0.988 − 0.149i)17-s + (0.5 + 0.866i)19-s + (0.988 + 0.149i)22-s + (0.365 + 0.930i)23-s + (−0.988 − 0.149i)26-s + (0.988 − 0.149i)29-s − 31-s + (0.623 + 0.781i)32-s + (−0.826 + 0.563i)34-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (−0.222 + 0.974i)8-s + (−0.826 − 0.563i)11-s + (0.826 + 0.563i)13-s + (−0.222 − 0.974i)16-s + (0.988 − 0.149i)17-s + (0.5 + 0.866i)19-s + (0.988 + 0.149i)22-s + (0.365 + 0.930i)23-s + (−0.988 − 0.149i)26-s + (0.988 − 0.149i)29-s − 31-s + (0.623 + 0.781i)32-s + (−0.826 + 0.563i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.371 + 0.928i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (194, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.371 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8451188024 + 0.5717934216i\)
\(L(\frac12)\) \(\approx\) \(0.8451188024 + 0.5717934216i\)
\(L(1)\) \(\approx\) \(0.7346294304 + 0.1831169759i\)
\(L(1)\) \(\approx\) \(0.7346294304 + 0.1831169759i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.900 + 0.433i)T \)
11 \( 1 + (-0.826 - 0.563i)T \)
13 \( 1 + (0.826 + 0.563i)T \)
17 \( 1 + (0.988 - 0.149i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.365 + 0.930i)T \)
29 \( 1 + (0.988 - 0.149i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.365 + 0.930i)T \)
41 \( 1 + (0.955 - 0.294i)T \)
43 \( 1 + (-0.955 - 0.294i)T \)
47 \( 1 + (0.900 - 0.433i)T \)
53 \( 1 + (0.365 + 0.930i)T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + (-0.623 - 0.781i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.623 + 0.781i)T \)
73 \( 1 + (0.826 - 0.563i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.826 + 0.563i)T \)
89 \( 1 + (0.0747 + 0.997i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−19.65060217183664321803741893257, −18.76846179976053316185170747381, −18.034559192651767256452669359855, −17.8288576683899712762048842707, −16.71267092365281091127262542913, −16.152188237924995152723078813, −15.474531047455255616881802992278, −14.69690399522236446968675554368, −13.58051690155903785396998834856, −12.81282856091683824243406330855, −12.30653650968657807584493558912, −11.36252715093118141554556430527, −10.588167444029150954878079846462, −10.20081971593639321356699271070, −9.20551120624020080731307187972, −8.57877208547965664618582383923, −7.72200833344391369657520774779, −7.20339115060072255129099484942, −6.19199761708474373025265696302, −5.27942973474579039012736668648, −4.23388187895046202282107255487, −3.17857615648618431004680336076, −2.62292695114488851590545699885, −1.51614505777129478040642700939, −0.58219251085166824781482557715, 0.946214031599860856787330901285, 1.71336874729386945035548730542, 2.8823237099376750211445250209, 3.68987225769000043219977516678, 5.08892653688006632168333919991, 5.67810090981418215768984800849, 6.41993193794904740901076593663, 7.403692591397690975543814678641, 7.9626787642067169199688615516, 8.71046404604722994815984327680, 9.47511129298753528811063419013, 10.246514327336850068249801359680, 10.88826778244799047645514662983, 11.65152287804743359022097434942, 12.41221331786702213336504201035, 13.68838548030476752367081734550, 13.99969890610769495561816193876, 15.044755803331282506851585607680, 15.73053963763911340459535600013, 16.35656585738145543632241099119, 16.864328547107546287338587488795, 17.79919111367775676930551804472, 18.59814300726489643962035929509, 18.78384522398689183397800741445, 19.69497566421479597082810854387

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