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Properties

Label 1-1575-1575.1181-r0-0-0
Degree $1$
Conductor $1575$
Sign $-0.564 + 0.825i$
Analytic cond. $7.31426$
Root an. cond. $7.31426$
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  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)8-s + (0.809 − 0.587i)11-s + (−0.913 + 0.406i)13-s + (0.913 − 0.406i)16-s + (−0.978 − 0.207i)17-s + (−0.669 + 0.743i)19-s + (0.669 + 0.743i)22-s + (0.809 − 0.587i)23-s + (−0.5 − 0.866i)26-s + (0.978 − 0.207i)29-s + (−0.669 + 0.743i)31-s + (0.5 + 0.866i)32-s + (0.104 − 0.994i)34-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (−0.978 + 0.207i)4-s + (−0.309 − 0.951i)8-s + (0.809 − 0.587i)11-s + (−0.913 + 0.406i)13-s + (0.913 − 0.406i)16-s + (−0.978 − 0.207i)17-s + (−0.669 + 0.743i)19-s + (0.669 + 0.743i)22-s + (0.809 − 0.587i)23-s + (−0.5 − 0.866i)26-s + (0.978 − 0.207i)29-s + (−0.669 + 0.743i)31-s + (0.5 + 0.866i)32-s + (0.104 − 0.994i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-0.564 + 0.825i$
Analytic conductor: \(7.31426\)
Root analytic conductor: \(7.31426\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1575} (1181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1575,\ (0:\ ),\ -0.564 + 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5458335073 + 1.033903298i\)
\(L(\frac12)\) \(\approx\) \(0.5458335073 + 1.033903298i\)
\(L(1)\) \(\approx\) \(0.8065610478 + 0.5039685565i\)
\(L(1)\) \(\approx\) \(0.8065610478 + 0.5039685565i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.104 + 0.994i)T \)
11 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (-0.913 + 0.406i)T \)
17 \( 1 + (-0.978 - 0.207i)T \)
19 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (0.978 - 0.207i)T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
41 \( 1 + (0.913 - 0.406i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.669 + 0.743i)T \)
53 \( 1 + (-0.669 - 0.743i)T \)
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 \( 1 + (0.104 + 0.994i)T \)
79 \( 1 + (0.669 + 0.743i)T \)
83 \( 1 + (0.669 - 0.743i)T \)
89 \( 1 + (0.913 + 0.406i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−20.02188831412717096672146311725, −19.70643510888991578744894104930, −19.06004138897316225351533329270, −17.93403169101925801619313082203, −17.49852853339263300204769684050, −16.85799986505507953080644934821, −15.513626298376881085091979505949, −14.85261783664059133203035473301, −14.20521820321231001348933131879, −13.170045414002693558359519064442, −12.7075263926681684677771509136, −11.89863148024316968599347183556, −11.10910425693987739288458473103, −10.501856573334611129121219709576, −9.397677848565287106247899598405, −9.15931903924147051287596258754, −8.037778663172511608071068376488, −7.060786526973466571199860913084, −6.1243112077202929305231025508, −4.94291647989046585839300505241, −4.463545059517593708700644106872, −3.478453143093260734061934939038, −2.48680264777086342307839016053, −1.78890759917340662433207799536, −0.51491088285427346451237176111, 0.95850798879150419147158751462, 2.371085673069310284086564306510, 3.53767661319960189006404728632, 4.40241768157757648448521375911, 5.06470776107451256695215667860, 6.17043235066000186579305559062, 6.70909905773639417795975777073, 7.463631658741093127645390842724, 8.57964590469496521296524804251, 8.92758264190820316790753288570, 9.88890371073949346160232317666, 10.78775327194585003471920450125, 11.87829206436064994782174229508, 12.55739401764720885492922458338, 13.42119659597395445222724731775, 14.25679582730038983050644556738, 14.660311750835907907123185593703, 15.56609927456817971338850912078, 16.31792352057303459025703826528, 17.03852410290257571579464852099, 17.46857337896473413129818383761, 18.495377266070313519165941775852, 19.14899648476090784685526321462, 19.8133242258809112238959388663, 20.97157303530381919078880142987

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