Properties

Label 2-3525-1.1-c1-0-111
Degree $2$
Conductor $3525$
Sign $-1$
Analytic cond. $28.1472$
Root an. cond. $5.30539$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2.27·2-s + 3-s + 3.15·4-s − 2.27·6-s + 0.797·7-s − 2.63·8-s + 9-s + 2.66·11-s + 3.15·12-s − 3.32·13-s − 1.81·14-s − 0.338·16-s − 2.55·17-s − 2.27·18-s + 4.89·19-s + 0.797·21-s − 6.06·22-s − 3.96·23-s − 2.63·24-s + 7.55·26-s + 27-s + 2.52·28-s − 7.42·29-s − 4.01·31-s + 6.03·32-s + 2.66·33-s + 5.80·34-s + ⋯
L(s)  = 1  − 1.60·2-s + 0.577·3-s + 1.57·4-s − 0.927·6-s + 0.301·7-s − 0.930·8-s + 0.333·9-s + 0.804·11-s + 0.911·12-s − 0.922·13-s − 0.484·14-s − 0.0847·16-s − 0.620·17-s − 0.535·18-s + 1.12·19-s + 0.174·21-s − 1.29·22-s − 0.827·23-s − 0.537·24-s + 1.48·26-s + 0.192·27-s + 0.476·28-s − 1.37·29-s − 0.721·31-s + 1.06·32-s + 0.464·33-s + 0.996·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3525\)    =    \(3 \cdot 5^{2} \cdot 47\)
Sign: $-1$
Analytic conductor: \(28.1472\)
Root analytic conductor: \(5.30539\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3525,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
47 \( 1 - T \)
good2 \( 1 + 2.27T + 2T^{2} \)
7 \( 1 - 0.797T + 7T^{2} \)
11 \( 1 - 2.66T + 11T^{2} \)
13 \( 1 + 3.32T + 13T^{2} \)
17 \( 1 + 2.55T + 17T^{2} \)
19 \( 1 - 4.89T + 19T^{2} \)
23 \( 1 + 3.96T + 23T^{2} \)
29 \( 1 + 7.42T + 29T^{2} \)
31 \( 1 + 4.01T + 31T^{2} \)
37 \( 1 - 10.0T + 37T^{2} \)
41 \( 1 + 8.81T + 41T^{2} \)
43 \( 1 + 6.51T + 43T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 + 7.40T + 59T^{2} \)
61 \( 1 - 0.913T + 61T^{2} \)
67 \( 1 + 5.01T + 67T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 - 7.63T + 73T^{2} \)
79 \( 1 + 2.38T + 79T^{2} \)
83 \( 1 - 2.19T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 - 11.5T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.122065678628273840832408099124, −7.80094391083951985522768829002, −7.04255279608272917265515866732, −6.39125169798531997319882171739, −5.19507726226053125909902174399, −4.23350117505392028683980169342, −3.17941280269483484402300124961, −2.09213612539067359217404046582, −1.44060144039195679427993956419, 0, 1.44060144039195679427993956419, 2.09213612539067359217404046582, 3.17941280269483484402300124961, 4.23350117505392028683980169342, 5.19507726226053125909902174399, 6.39125169798531997319882171739, 7.04255279608272917265515866732, 7.80094391083951985522768829002, 8.122065678628273840832408099124

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