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Properties

Label 1-1925-1925.1094-r0-0-0
Degree $1$
Conductor $1925$
Sign $-0.823 - 0.567i$
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  + (0.978 − 0.207i)2-s + (0.5 − 0.866i)3-s + (0.913 − 0.406i)4-s + (0.309 − 0.951i)6-s + (0.809 − 0.587i)8-s + (−0.5 − 0.866i)9-s + (0.104 − 0.994i)12-s − 13-s + (0.669 − 0.743i)16-s + (−0.669 − 0.743i)17-s + (−0.669 − 0.743i)18-s + (0.913 + 0.406i)19-s + (−0.913 − 0.406i)23-s + (−0.104 − 0.994i)24-s + (−0.978 + 0.207i)26-s − 27-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)2-s + (0.5 − 0.866i)3-s + (0.913 − 0.406i)4-s + (0.309 − 0.951i)6-s + (0.809 − 0.587i)8-s + (−0.5 − 0.866i)9-s + (0.104 − 0.994i)12-s − 13-s + (0.669 − 0.743i)16-s + (−0.669 − 0.743i)17-s + (−0.669 − 0.743i)18-s + (0.913 + 0.406i)19-s + (−0.913 − 0.406i)23-s + (−0.104 − 0.994i)24-s + (−0.978 + 0.207i)26-s − 27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.823 - 0.567i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.823 - 0.567i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1094, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ -0.823 - 0.567i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9093916337 - 2.918886164i\)
\(L(\frac12)\) \(\approx\) \(0.9093916337 - 2.918886164i\)
\(L(1)\) \(\approx\) \(1.640005480 - 1.174362850i\)
\(L(1)\) \(\approx\) \(1.640005480 - 1.174362850i\)

Euler product

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

−20.24049314958831105558624492386, −19.977358340413153138644931682045, −19.29309393941419509561510363503, −17.9751421753512089487114647847, −17.099013809594418442699802789023, −16.488681918051732572076629351440, −15.70868890000069179537743583436, −15.12413421933275234806142903042, −14.5892329569459483252622082288, −13.71162766181338438966501275805, −13.285643535576017944312138634835, −12.13587257548621847477187128326, −11.60461382502916409088345515884, −10.61462029724071047770291118927, −10.04091216075323676162720539473, −9.069431723469779653938627385161, −8.1840823298745914742494260857, −7.444899366357443649398883376474, −6.57361053452853827045892727479, −5.50148018749939401777578017985, −4.944489015708481615759880079124, −4.13734151497101554829039626782, −3.390728337411024997230009154012, −2.58824701113956681960503817489, −1.74268864054243259003774711413, 0.615317674816055835977507427342, 1.909329833251040901247360709721, 2.42364842802628664659033833329, 3.31669418557774963236353239705, 4.19087711417985567961582489542, 5.17515510865485819441583048448, 5.9654158426430839430009081960, 6.84039100707418299499571997161, 7.42557571170887174980788291484, 8.16435865115212345424656780594, 9.368641226969922248739797359958, 9.9984462514779066477431594972, 11.13250722972680566513802233687, 11.93178377391651429864564110502, 12.31209756948243737256468905139, 13.16800125096591394629775653843, 13.95667634868471200800263425114, 14.218385240541134511316674331705, 15.210683091249494833784086042264, 15.77765266800529254793428814127, 16.80434272270750815781038542119, 17.61000418323949176419841011232, 18.5219916399930814215823637323, 19.15023931238909741636685245699, 19.92087460895153112823681220232

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