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Properties

Label 1-1925-1925.1033-r0-0-0
Degree $1$
Conductor $1925$
Sign $0.103 - 0.994i$
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  + (0.994 − 0.104i)2-s + (−0.207 − 0.978i)3-s + (0.978 − 0.207i)4-s + (−0.309 − 0.951i)6-s + (0.951 − 0.309i)8-s + (−0.913 + 0.406i)9-s + (−0.406 − 0.913i)12-s + (−0.587 + 0.809i)13-s + (0.913 − 0.406i)16-s + (0.743 + 0.669i)17-s + (−0.866 + 0.5i)18-s + (−0.978 − 0.207i)19-s + (0.994 − 0.104i)23-s + (−0.5 − 0.866i)24-s + (−0.5 + 0.866i)26-s + (0.587 + 0.809i)27-s + ⋯
L(s)  = 1  + (0.994 − 0.104i)2-s + (−0.207 − 0.978i)3-s + (0.978 − 0.207i)4-s + (−0.309 − 0.951i)6-s + (0.951 − 0.309i)8-s + (−0.913 + 0.406i)9-s + (−0.406 − 0.913i)12-s + (−0.587 + 0.809i)13-s + (0.913 − 0.406i)16-s + (0.743 + 0.669i)17-s + (−0.866 + 0.5i)18-s + (−0.978 − 0.207i)19-s + (0.994 − 0.104i)23-s + (−0.5 − 0.866i)24-s + (−0.5 + 0.866i)26-s + (0.587 + 0.809i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.103 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.103 - 0.994i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1033, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ 0.103 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.164386496 - 1.949887336i\)
\(L(\frac12)\) \(\approx\) \(2.164386496 - 1.949887336i\)
\(L(1)\) \(\approx\) \(1.686199655 - 0.7351479766i\)
\(L(1)\) \(\approx\) \(1.686199655 - 0.7351479766i\)

Euler product

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

−20.53806031850539937815217403991, −19.74319420899418285608077445158, −18.99463621724670069196213517271, −17.67023259405825577288968033367, −17.07972430196664508751888039144, −16.379015939889594804868352240921, −15.6884626150529680468995135075, −15.03811428359157793929279621493, −14.461114717614812930135059180117, −13.77457618789232237882719067328, −12.6564488450134369508455237544, −12.23225164509372029696359350782, −11.283731230714105944816689931664, −10.60336892158621059427183616862, −10.03562490147658351129951520289, −8.9891041005939561018672654763, −8.07137642883569443017714125761, −7.15837749079534678568893520884, −6.26648443634155396035648418574, −5.41041326489041928883686294594, −4.88513455408097688694693903841, −4.12812932671802790783432033706, −3.072713324541942694174306557716, −2.7252919434405853086593908029, −1.14431321301914929427559972403, 0.79649706959103490840490806954, 1.979709995228307225986144333950, 2.44284503063718747404194193427, 3.57661537441738653281939759967, 4.50175240724650657140987058759, 5.34503544608967806790919775481, 6.18968497848497632093865723952, 6.74478451249524153927410621979, 7.536932174357441833159607599302, 8.28234451768059746560570057011, 9.407205785649753792070783799572, 10.52878468369261817241075769672, 11.14375506085641221218862735437, 12.06201892316098629410320954108, 12.380364520275232707319269105067, 13.2778514694882312317396790050, 13.77808336755639310811970203329, 14.68823592876277779396002280109, 15.09561469353460013160922677769, 16.2969307489255705687004235584, 16.96484718310403906675857722934, 17.46901626748918915069155289063, 18.74366971320351347953461143355, 19.28434553585982859258396964900, 19.634146331213860281713786508468

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