Properties

Label 2-4205-1.1-c1-0-109
Degree $2$
Conductor $4205$
Sign $-1$
Analytic cond. $33.5770$
Root an. cond. $5.79457$
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  − 1.53·2-s − 1.70·3-s + 0.369·4-s − 5-s + 2.63·6-s + 0.630·7-s + 2.51·8-s − 0.0783·9-s + 1.53·10-s − 0.290·11-s − 0.630·12-s − 0.921·13-s − 0.971·14-s + 1.70·15-s − 4.60·16-s − 4.97·17-s + 0.120·18-s + 6.04·19-s − 0.369·20-s − 1.07·21-s + 0.447·22-s + 2.29·23-s − 4.29·24-s + 25-s + 1.41·26-s + 5.26·27-s + 0.232·28-s + ⋯
L(s)  = 1  − 1.08·2-s − 0.986·3-s + 0.184·4-s − 0.447·5-s + 1.07·6-s + 0.238·7-s + 0.887·8-s − 0.0261·9-s + 0.486·10-s − 0.0876·11-s − 0.182·12-s − 0.255·13-s − 0.259·14-s + 0.441·15-s − 1.15·16-s − 1.20·17-s + 0.0284·18-s + 1.38·19-s − 0.0825·20-s − 0.235·21-s + 0.0954·22-s + 0.477·23-s − 0.875·24-s + 0.200·25-s + 0.278·26-s + 1.01·27-s + 0.0440·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4205\)    =    \(5 \cdot 29^{2}\)
Sign: $-1$
Analytic conductor: \(33.5770\)
Root analytic conductor: \(5.79457\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4205,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
29 \( 1 \)
good2 \( 1 + 1.53T + 2T^{2} \)
3 \( 1 + 1.70T + 3T^{2} \)
7 \( 1 - 0.630T + 7T^{2} \)
11 \( 1 + 0.290T + 11T^{2} \)
13 \( 1 + 0.921T + 13T^{2} \)
17 \( 1 + 4.97T + 17T^{2} \)
19 \( 1 - 6.04T + 19T^{2} \)
23 \( 1 - 2.29T + 23T^{2} \)
31 \( 1 + 10.0T + 31T^{2} \)
37 \( 1 + 1.55T + 37T^{2} \)
41 \( 1 + 0.340T + 41T^{2} \)
43 \( 1 - 5.70T + 43T^{2} \)
47 \( 1 - 1.12T + 47T^{2} \)
53 \( 1 + 0.340T + 53T^{2} \)
59 \( 1 - 9.75T + 59T^{2} \)
61 \( 1 + 3.07T + 61T^{2} \)
67 \( 1 + 5.70T + 67T^{2} \)
71 \( 1 - 9.07T + 71T^{2} \)
73 \( 1 - 6.94T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 2.78T + 83T^{2} \)
89 \( 1 + 4.73T + 89T^{2} \)
97 \( 1 - 15.8T + 97T^{2} \)
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

−8.096385384848365631121748608495, −7.31044540342434022547532448356, −6.89648402378590384313482113487, −5.79832512538582901170432416220, −5.07899338481445844131392915514, −4.47642995867971452296545149794, −3.38046739438629461804665039384, −2.09627536665805868261187832212, −0.927227859925192139787391932746, 0, 0.927227859925192139787391932746, 2.09627536665805868261187832212, 3.38046739438629461804665039384, 4.47642995867971452296545149794, 5.07899338481445844131392915514, 5.79832512538582901170432416220, 6.89648402378590384313482113487, 7.31044540342434022547532448356, 8.096385384848365631121748608495

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