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Properties

Label 1-1925-1925.479-r0-0-0
Degree $1$
Conductor $1925$
Sign $-0.755 - 0.655i$
Analytic cond. $8.93966$
Root an. cond. $8.93966$
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)2-s + (0.913 + 0.406i)3-s + (−0.5 − 0.866i)4-s + (−0.809 + 0.587i)6-s + 8-s + (0.669 + 0.743i)9-s + (−0.104 − 0.994i)12-s + (−0.309 + 0.951i)13-s + (−0.5 + 0.866i)16-s + (−0.913 − 0.406i)17-s + (−0.978 + 0.207i)18-s + (−0.5 + 0.866i)19-s + (−0.913 + 0.406i)23-s + (0.913 + 0.406i)24-s + (−0.669 − 0.743i)26-s + (0.309 + 0.951i)27-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.913 + 0.406i)3-s + (−0.5 − 0.866i)4-s + (−0.809 + 0.587i)6-s + 8-s + (0.669 + 0.743i)9-s + (−0.104 − 0.994i)12-s + (−0.309 + 0.951i)13-s + (−0.5 + 0.866i)16-s + (−0.913 − 0.406i)17-s + (−0.978 + 0.207i)18-s + (−0.5 + 0.866i)19-s + (−0.913 + 0.406i)23-s + (0.913 + 0.406i)24-s + (−0.669 − 0.743i)26-s + (0.309 + 0.951i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.755 - 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.755 - 0.655i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ -0.755 - 0.655i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2215155772 + 0.5937027499i\)
\(L(\frac12)\) \(\approx\) \(-0.2215155772 + 0.5937027499i\)
\(L(1)\) \(\approx\) \(0.6797403215 + 0.5226753619i\)
\(L(1)\) \(\approx\) \(0.6797403215 + 0.5226753619i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.913 - 0.406i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.913 + 0.406i)T \)
29 \( 1 - T \)
31 \( 1 + (-0.669 + 0.743i)T \)
37 \( 1 + (0.978 + 0.207i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.104 - 0.994i)T \)
53 \( 1 + (0.104 - 0.994i)T \)
59 \( 1 + (-0.669 + 0.743i)T \)
61 \( 1 + (-0.978 - 0.207i)T \)
67 \( 1 + (0.978 - 0.207i)T \)
71 \( 1 + (0.309 - 0.951i)T \)
73 \( 1 + (-0.913 - 0.406i)T \)
79 \( 1 + (-0.913 + 0.406i)T \)
83 \( 1 + (0.809 - 0.587i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−19.805719214460183378189137118875, −18.91718903207320010690756523730, −18.316808899362974616269428671683, −17.641256071066224611414271479619, −16.98858909166521264902487070333, −15.862899849293856235363381148318, −15.12032557718904858114143889479, −14.33558629957790142128138701916, −13.381740899044083975882854372352, −12.92774112518892458541486019540, −12.35827925569931694442523142708, −11.27946448412153914008206243020, −10.65752985482586676474849267680, −9.69015174918807363079467315808, −9.17222722008446679313190454636, −8.31959044115305510122740657422, −7.774601461188078425625848850975, −6.984385172923220666049913670896, −5.9006857080751437611645795745, −4.50416898645286954681945288656, −3.93448824550508949905126937284, −2.857068847841016245403702367579, −2.35248442025200957397857701339, −1.43742306480214309356684761047, −0.213932710284022218865554585712, 1.64078609068317193828367181811, 2.2172115546997203319971817505, 3.66941837522649324736090026309, 4.32575074368266234115892553909, 5.15613438869051944493723720386, 6.16249509207733614312133135973, 7.060353299763362873612459097679, 7.67631476922855066559959771726, 8.52854916479387596509773166355, 9.13367541024850726554109437755, 9.74448614611890974783938819225, 10.50766446898694743854220805173, 11.354900798092309327497597275887, 12.55578606219032750098556416694, 13.5525341004377972348131989017, 13.99900640796929859729150465385, 14.79019688854526635109028112971, 15.27893343240267717261434354065, 16.23705241770827500625469022803, 16.53051313930444338461158099199, 17.51367989644577504029431416513, 18.440393378110951747088999994753, 18.87651533574405988740733447878, 19.793394988396867153457534455498, 20.166942975020666584905544099165

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