↑

Properties

Label 2-2007-223.215-c0-0-0
Degree $2$
Conductor $2007$
Sign $-0.929 + 0.369i$
Analytic cond. $1.00162$
Root an. cond. $1.00081$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.524 + 0.851i)4-s + (−0.745 + 0.504i)7-s + (−1.95 − 0.420i)13-s + (−0.450 − 0.892i)16-s + (0.157 + 0.390i)19-s + (−0.996 + 0.0848i)25-s + (−0.0382 − 0.899i)28-s + (−0.183 − 1.43i)31-s + (−0.167 + 0.190i)37-s + (−0.743 − 0.0632i)43-s + (−0.0705 + 0.175i)49-s + (1.38 − 1.44i)52-s + (−1.09 − 0.235i)61-s + (0.996 + 0.0848i)64-s + (−0.567 − 0.349i)67-s + ⋯
L(s)  = 1  + (−0.524 + 0.851i)4-s + (−0.745 + 0.504i)7-s + (−1.95 − 0.420i)13-s + (−0.450 − 0.892i)16-s + (0.157 + 0.390i)19-s + (−0.996 + 0.0848i)25-s + (−0.0382 − 0.899i)28-s + (−0.183 − 1.43i)31-s + (−0.167 + 0.190i)37-s + (−0.743 − 0.0632i)43-s + (−0.0705 + 0.175i)49-s + (1.38 − 1.44i)52-s + (−1.09 − 0.235i)61-s + (0.996 + 0.0848i)64-s + (−0.567 − 0.349i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2007\)    =    \(3^{2} \cdot 223\)
Sign: $-0.929 + 0.369i$
Analytic conductor: \(1.00162\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2007} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2007,\ (\ :0),\ -0.929 + 0.369i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1099624307\)
\(L(\frac12)\) \(\approx\) \(0.1099624307\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
223 \( 1 + (-0.996 + 0.0848i)T \)
good2 \( 1 + (0.524 - 0.851i)T^{2} \)
5 \( 1 + (0.996 - 0.0848i)T^{2} \)
7 \( 1 + (0.745 - 0.504i)T + (0.372 - 0.927i)T^{2} \)
11 \( 1 + (0.127 - 0.991i)T^{2} \)
13 \( 1 + (1.95 + 0.420i)T + (0.911 + 0.411i)T^{2} \)
17 \( 1 + (-0.127 + 0.991i)T^{2} \)
19 \( 1 + (-0.157 - 0.390i)T + (-0.721 + 0.691i)T^{2} \)
23 \( 1 + (0.996 - 0.0848i)T^{2} \)
29 \( 1 + (-0.594 + 0.803i)T^{2} \)
31 \( 1 + (0.183 + 1.43i)T + (-0.967 + 0.251i)T^{2} \)
37 \( 1 + (0.167 - 0.190i)T + (-0.127 - 0.991i)T^{2} \)
41 \( 1 + (0.210 + 0.977i)T^{2} \)
43 \( 1 + (0.743 + 0.0632i)T + (0.985 + 0.169i)T^{2} \)
47 \( 1 + (-0.721 - 0.691i)T^{2} \)
53 \( 1 + (-0.828 - 0.559i)T^{2} \)
59 \( 1 + (-0.778 + 0.628i)T^{2} \)
61 \( 1 + (1.09 + 0.235i)T + (0.911 + 0.411i)T^{2} \)
67 \( 1 + (0.567 + 0.349i)T + (0.450 + 0.892i)T^{2} \)
71 \( 1 + (-0.778 + 0.628i)T^{2} \)
73 \( 1 + (-0.757 - 0.725i)T + (0.0424 + 0.999i)T^{2} \)
79 \( 1 + (-0.106 + 0.131i)T + (-0.210 - 0.977i)T^{2} \)
83 \( 1 + (0.985 - 0.169i)T^{2} \)
89 \( 1 + (-0.996 - 0.0848i)T^{2} \)
97 \( 1 + (1.25 + 0.0533i)T + (0.996 + 0.0848i)T^{2} \)
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

−9.588605935730679906704797418911, −9.201493857776866673041380657267, −8.031955135574306077776012813064, −7.65602954378295048746194082395, −6.75956804575730348761064662477, −5.72245374151996532560427039714, −4.92811915734078758656421011062, −4.00404072736844531831331480361, −3.05522796186797145349440208676, −2.28808801099254396404029779315, 0.07440364083635765724015653367, 1.73488518434287462499817633931, 2.94178445785780530559437080929, 4.11761750136804257766927069684, 4.88378324493784557145601053927, 5.59267990374521942702759017632, 6.67755182043776666118248305483, 7.13235276342662177542833920430, 8.165095910674338865170302611140, 9.231616342225652328753370462651

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