Properties

Label 2-4235-1.1-c1-0-131
Degree $2$
Conductor $4235$
Sign $-1$
Analytic cond. $33.8166$
Root an. cond. $5.81520$
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  − 1.35·2-s − 0.242·3-s − 0.159·4-s + 5-s + 0.329·6-s − 7-s + 2.92·8-s − 2.94·9-s − 1.35·10-s + 0.0386·12-s − 2.34·13-s + 1.35·14-s − 0.242·15-s − 3.65·16-s − 0.927·17-s + 3.99·18-s + 1.95·19-s − 0.159·20-s + 0.242·21-s + 2.68·23-s − 0.711·24-s + 25-s + 3.18·26-s + 1.44·27-s + 0.159·28-s + 0.245·29-s + 0.329·30-s + ⋯
L(s)  = 1  − 0.959·2-s − 0.140·3-s − 0.0795·4-s + 0.447·5-s + 0.134·6-s − 0.377·7-s + 1.03·8-s − 0.980·9-s − 0.429·10-s + 0.0111·12-s − 0.650·13-s + 0.362·14-s − 0.0627·15-s − 0.914·16-s − 0.224·17-s + 0.940·18-s + 0.448·19-s − 0.0355·20-s + 0.0530·21-s + 0.559·23-s − 0.145·24-s + 0.200·25-s + 0.624·26-s + 0.277·27-s + 0.0300·28-s + 0.0456·29-s + 0.0601·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4235\)    =    \(5 \cdot 7 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(33.8166\)
Root analytic conductor: \(5.81520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4235} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4235,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + 1.35T + 2T^{2} \)
3 \( 1 + 0.242T + 3T^{2} \)
13 \( 1 + 2.34T + 13T^{2} \)
17 \( 1 + 0.927T + 17T^{2} \)
19 \( 1 - 1.95T + 19T^{2} \)
23 \( 1 - 2.68T + 23T^{2} \)
29 \( 1 - 0.245T + 29T^{2} \)
31 \( 1 - 2.99T + 31T^{2} \)
37 \( 1 - 6.23T + 37T^{2} \)
41 \( 1 - 6.05T + 41T^{2} \)
43 \( 1 + 9.64T + 43T^{2} \)
47 \( 1 - 5.84T + 47T^{2} \)
53 \( 1 + 4.58T + 53T^{2} \)
59 \( 1 + 9.66T + 59T^{2} \)
61 \( 1 + 2.82T + 61T^{2} \)
67 \( 1 - 15.0T + 67T^{2} \)
71 \( 1 + 2.21T + 71T^{2} \)
73 \( 1 + 0.100T + 73T^{2} \)
79 \( 1 + 6.31T + 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 - 2.73T + 89T^{2} \)
97 \( 1 - 3.11T + 97T^{2} \)
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

−8.194941677398240549675313750463, −7.45987668873723744552414132907, −6.69661782997850585368351299552, −5.87382993973365391984538623364, −5.10339136200565389193831863709, −4.36850659751532971765998604641, −3.14674602602346814070299842064, −2.34263378196145346687608772081, −1.10975260985523699320964583679, 0, 1.10975260985523699320964583679, 2.34263378196145346687608772081, 3.14674602602346814070299842064, 4.36850659751532971765998604641, 5.10339136200565389193831863709, 5.87382993973365391984538623364, 6.69661782997850585368351299552, 7.45987668873723744552414132907, 8.194941677398240549675313750463

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