Properties

Label 2-4005-1.1-c1-0-109
Degree $2$
Conductor $4005$
Sign $-1$
Analytic cond. $31.9800$
Root an. cond. $5.65509$
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  − 2.24·2-s + 3.03·4-s + 5-s + 1.09·7-s − 2.31·8-s − 2.24·10-s + 3.76·11-s − 6.02·13-s − 2.46·14-s − 0.869·16-s − 3.14·17-s + 0.176·19-s + 3.03·20-s − 8.45·22-s + 0.808·23-s + 25-s + 13.5·26-s + 3.33·28-s + 5.42·29-s + 3.31·31-s + 6.58·32-s + 7.05·34-s + 1.09·35-s − 11.5·37-s − 0.396·38-s − 2.31·40-s + 1.30·41-s + ⋯
L(s)  = 1  − 1.58·2-s + 1.51·4-s + 0.447·5-s + 0.415·7-s − 0.818·8-s − 0.709·10-s + 1.13·11-s − 1.66·13-s − 0.658·14-s − 0.217·16-s − 0.762·17-s + 0.0405·19-s + 0.678·20-s − 1.80·22-s + 0.168·23-s + 0.200·25-s + 2.64·26-s + 0.629·28-s + 1.00·29-s + 0.596·31-s + 1.16·32-s + 1.21·34-s + 0.185·35-s − 1.89·37-s − 0.0643·38-s − 0.366·40-s + 0.203·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4005 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4005\)    =    \(3^{2} \cdot 5 \cdot 89\)
Sign: $-1$
Analytic conductor: \(31.9800\)
Root analytic conductor: \(5.65509\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4005,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
89 \( 1 - T \)
good2 \( 1 + 2.24T + 2T^{2} \)
7 \( 1 - 1.09T + 7T^{2} \)
11 \( 1 - 3.76T + 11T^{2} \)
13 \( 1 + 6.02T + 13T^{2} \)
17 \( 1 + 3.14T + 17T^{2} \)
19 \( 1 - 0.176T + 19T^{2} \)
23 \( 1 - 0.808T + 23T^{2} \)
29 \( 1 - 5.42T + 29T^{2} \)
31 \( 1 - 3.31T + 31T^{2} \)
37 \( 1 + 11.5T + 37T^{2} \)
41 \( 1 - 1.30T + 41T^{2} \)
43 \( 1 + 0.0238T + 43T^{2} \)
47 \( 1 - 2.23T + 47T^{2} \)
53 \( 1 + 7.11T + 53T^{2} \)
59 \( 1 + 9.73T + 59T^{2} \)
61 \( 1 + 14.5T + 61T^{2} \)
67 \( 1 + 3.60T + 67T^{2} \)
71 \( 1 - 3.36T + 71T^{2} \)
73 \( 1 - 6.82T + 73T^{2} \)
79 \( 1 - 6.52T + 79T^{2} \)
83 \( 1 + 3.43T + 83T^{2} \)
97 \( 1 + 12.9T + 97T^{2} \)
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

−8.222709363981938233751846754941, −7.48915694547118141034218280020, −6.80932443859109450505083829528, −6.31229630420273991990681051960, −5.01316841405774376335772106538, −4.44321735584031583055367662018, −2.99206078025851613994106614627, −2.06277500183195424395542568159, −1.32797050202936110129523365619, 0, 1.32797050202936110129523365619, 2.06277500183195424395542568159, 2.99206078025851613994106614627, 4.44321735584031583055367662018, 5.01316841405774376335772106538, 6.31229630420273991990681051960, 6.80932443859109450505083829528, 7.48915694547118141034218280020, 8.222709363981938233751846754941

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