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Properties

Label 1-1425-1425.944-r0-0-0
Degree $1$
Conductor $1425$
Sign $0.902 + 0.431i$
Analytic cond. $6.61767$
Root an. cond. $6.61767$
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  + (−0.438 − 0.898i)2-s + (−0.615 + 0.788i)4-s + (0.5 + 0.866i)7-s + (0.978 + 0.207i)8-s + (0.104 + 0.994i)11-s + (−0.997 + 0.0697i)13-s + (0.559 − 0.829i)14-s + (−0.241 − 0.970i)16-s + (0.990 − 0.139i)17-s + (0.848 − 0.529i)22-s + (0.961 − 0.275i)23-s + (0.5 + 0.866i)26-s + (−0.990 − 0.139i)28-s + (0.990 + 0.139i)29-s + (−0.669 + 0.743i)31-s + (−0.766 + 0.642i)32-s + ⋯
L(s)  = 1  + (−0.438 − 0.898i)2-s + (−0.615 + 0.788i)4-s + (0.5 + 0.866i)7-s + (0.978 + 0.207i)8-s + (0.104 + 0.994i)11-s + (−0.997 + 0.0697i)13-s + (0.559 − 0.829i)14-s + (−0.241 − 0.970i)16-s + (0.990 − 0.139i)17-s + (0.848 − 0.529i)22-s + (0.961 − 0.275i)23-s + (0.5 + 0.866i)26-s + (−0.990 − 0.139i)28-s + (0.990 + 0.139i)29-s + (−0.669 + 0.743i)31-s + (−0.766 + 0.642i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1425 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1425\)    =    \(3 \cdot 5^{2} \cdot 19\)
Sign: $0.902 + 0.431i$
Analytic conductor: \(6.61767\)
Root analytic conductor: \(6.61767\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1425} (944, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1425,\ (0:\ ),\ 0.902 + 0.431i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.120370167 + 0.2541573777i\)
\(L(\frac12)\) \(\approx\) \(1.120370167 + 0.2541573777i\)
\(L(1)\) \(\approx\) \(0.8711061064 - 0.1013253908i\)
\(L(1)\) \(\approx\) \(0.8711061064 - 0.1013253908i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.438 - 0.898i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.104 + 0.994i)T \)
13 \( 1 + (-0.997 + 0.0697i)T \)
17 \( 1 + (0.990 - 0.139i)T \)
23 \( 1 + (0.961 - 0.275i)T \)
29 \( 1 + (0.990 + 0.139i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.241 - 0.970i)T \)
43 \( 1 + (0.939 - 0.342i)T \)
47 \( 1 + (0.990 + 0.139i)T \)
53 \( 1 + (0.615 - 0.788i)T \)
59 \( 1 + (-0.719 + 0.694i)T \)
61 \( 1 + (0.961 - 0.275i)T \)
67 \( 1 + (-0.882 + 0.469i)T \)
71 \( 1 + (0.0348 + 0.999i)T \)
73 \( 1 + (0.997 + 0.0697i)T \)
79 \( 1 + (-0.848 + 0.529i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (-0.241 + 0.970i)T \)
97 \( 1 + (-0.882 - 0.469i)T \)
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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

−20.64265460961664877365960838671, −19.657558459053555041811836389536, −19.16045916664127731188485926326, −18.367159613990641765472334477103, −17.43743018759548517839045812457, −16.89104782069005031361922401402, −16.44552142062982364668045176418, −15.37592468185682354350848105320, −14.6699277059792379750262582807, −14.005987848718208804256652932049, −13.427777160127961344077334272297, −12.33917476131212484922193910137, −11.24594238580442013201621793782, −10.5243074345237506435733697996, −9.799387365069652449498493857580, −8.92830466809964310979784636352, −8.02154846691395644819927295263, −7.493467998206041121742353885616, −6.6806914564870438854833836643, −5.70282674798015779754911389295, −4.97098014162158248096432551687, −4.10454984323211457446348448265, −2.99877121385474908060464182668, −1.48824805007440435365626646080, −0.61553556748334968132954282476, 1.10366805858859786388772163446, 2.14112931895903444451307827068, 2.729132896804361458234749301030, 3.860720931601369237975600326242, 4.89769784787832858378674885588, 5.39239341979612876748751043062, 7.00653010789280469926930522867, 7.57024909095728724662125906893, 8.635269924239350452249260402307, 9.18717160968141492960012488812, 10.05222692636809403966432354503, 10.701231017458589403668957418413, 11.74968428395097359294651172429, 12.33532037206473571001707266070, 12.67563165477727184043905327567, 14.03606824157923876945946725967, 14.58179854786880317791661319992, 15.48555774649677353392309730342, 16.48178670725378946335842530326, 17.4105857889031274633711324, 17.75689278865066463942057508832, 18.7227478703966911973382178566, 19.22591906176689239818599857239, 20.04761956119596628591076103853, 20.859400030534865021792059071368

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