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Properties

Label 1-1925-1925.1059-r0-0-0
Degree $1$
Conductor $1925$
Sign $0.981 - 0.189i$
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 + (−0.669 − 0.743i)3-s + (−0.5 + 0.866i)4-s + (0.309 − 0.951i)6-s − 8-s + (−0.104 + 0.994i)9-s + (0.978 − 0.207i)12-s + (0.809 + 0.587i)13-s + (−0.5 − 0.866i)16-s + (−0.669 − 0.743i)17-s + (−0.913 + 0.406i)18-s + (−0.5 − 0.866i)19-s + (−0.669 + 0.743i)23-s + (0.669 + 0.743i)24-s + (−0.104 + 0.994i)26-s + (0.809 − 0.587i)27-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.669 − 0.743i)3-s + (−0.5 + 0.866i)4-s + (0.309 − 0.951i)6-s − 8-s + (−0.104 + 0.994i)9-s + (0.978 − 0.207i)12-s + (0.809 + 0.587i)13-s + (−0.5 − 0.866i)16-s + (−0.669 − 0.743i)17-s + (−0.913 + 0.406i)18-s + (−0.5 − 0.866i)19-s + (−0.669 + 0.743i)23-s + (0.669 + 0.743i)24-s + (−0.104 + 0.994i)26-s + (0.809 − 0.587i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $0.981 - 0.189i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1059, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ 0.981 - 0.189i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.131737342 - 0.1082863079i\)
\(L(\frac12)\) \(\approx\) \(1.131737342 - 0.1082863079i\)
\(L(1)\) \(\approx\) \(0.9294834906 + 0.2157511112i\)
\(L(1)\) \(\approx\) \(0.9294834906 + 0.2157511112i\)

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 \( 1 + (-0.669 - 0.743i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
17 \( 1 + (-0.669 - 0.743i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.669 + 0.743i)T \)
29 \( 1 + T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.913 - 0.406i)T \)
41 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 - T \)
47 \( 1 + (0.978 - 0.207i)T \)
53 \( 1 + (0.978 + 0.207i)T \)
59 \( 1 + (-0.104 - 0.994i)T \)
61 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (-0.913 + 0.406i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 + (-0.669 - 0.743i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (-0.309 + 0.951i)T \)
89 \( 1 + (0.669 - 0.743i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−20.30426573108815364466224968657, −19.51322472707843029771821013098, −18.608070031039309173045960328060, −17.90489617850451606753197759953, −17.31609557036580447103407951989, −16.25107713815941969075041318527, −15.60366418657696707745159223487, −14.91371214833022175340748669670, −14.159663633567428425469487064, −13.30080570783752608418312616921, −12.400023140417377876526890404, −12.00925722979724565790685883825, −11.00165935486906490901399705138, −10.38300847141249052040023795095, −10.14110593752994618788370049297, −8.79957976948962441860517469888, −8.477145625880635904674097366312, −6.761653652122438769020906747248, −6.04338604189228725709990136215, −5.43092981921495583398488781828, −4.45708337105327095274413771021, −3.886255536264030735045263011754, −3.1071539821885915935684756295, −1.94132264179122869733715126149, −0.88581617486853667760256626089, 0.4619827717946986672488133464, 1.87162985210645303564051860144, 2.85494480182170983860054015853, 4.07125706780724977602939596914, 4.76957036452392722789479777907, 5.63611336352789114919628772534, 6.379091007210057666030410882340, 6.92152863581147048201589552027, 7.67832053717813335109979382008, 8.54326451323675734598892246085, 9.22380669709250724279747990526, 10.45322756073619249089190176197, 11.53777076795974509695027088690, 11.78995925213938241980045538972, 12.85586632150394604902629500079, 13.536631017390564979479152663137, 13.82720787879053665289446094312, 14.92940533248431096676392002121, 15.8190053703033132289277129578, 16.25445399033144772832894810770, 17.115059422442588691496460354471, 17.76994380507432761808210866716, 18.2536532114972471655850049502, 19.079431560776105440474193512443, 19.946411012259157939695785445498

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