↑

Properties

Label 2-2007-223.189-c0-0-0
Degree $2$
Conductor $2007$
Sign $-0.331 - 0.943i$
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  + (0.292 + 0.956i)4-s + (0.0703 + 1.65i)7-s + (−0.660 − 0.0846i)13-s + (−0.828 + 0.559i)16-s + (0.253 + 0.0215i)19-s + (0.778 − 0.628i)25-s + (−1.56 + 0.552i)28-s + (−1.03 + 1.67i)31-s + (0.915 − 0.511i)37-s + (−1.55 − 1.25i)43-s + (−1.73 + 0.148i)49-s + (−0.112 − 0.656i)52-s + (1.98 + 0.253i)61-s + (−0.778 − 0.628i)64-s + (−0.787 + 0.241i)67-s + ⋯
L(s)  = 1  + (0.292 + 0.956i)4-s + (0.0703 + 1.65i)7-s + (−0.660 − 0.0846i)13-s + (−0.828 + 0.559i)16-s + (0.253 + 0.0215i)19-s + (0.778 − 0.628i)25-s + (−1.56 + 0.552i)28-s + (−1.03 + 1.67i)31-s + (0.915 − 0.511i)37-s + (−1.55 − 1.25i)43-s + (−1.73 + 0.148i)49-s + (−0.112 − 0.656i)52-s + (1.98 + 0.253i)61-s + (−0.778 − 0.628i)64-s + (−0.787 + 0.241i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2007\)    =    \(3^{2} \cdot 223\)
Sign: $-0.331 - 0.943i$
Analytic conductor: \(1.00162\)
Root analytic conductor: \(1.00081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2007} (1081, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2007,\ (\ :0),\ -0.331 - 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.120417931\)
\(L(\frac12)\) \(\approx\) \(1.120417931\)
\(L(1)\) 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 \( 1 + (0.778 - 0.628i)T \)
good2 \( 1 + (-0.292 - 0.956i)T^{2} \)
5 \( 1 + (-0.778 + 0.628i)T^{2} \)
7 \( 1 + (-0.0703 - 1.65i)T + (-0.996 + 0.0848i)T^{2} \)
11 \( 1 + (-0.524 - 0.851i)T^{2} \)
13 \( 1 + (0.660 + 0.0846i)T + (0.967 + 0.251i)T^{2} \)
17 \( 1 + (0.524 + 0.851i)T^{2} \)
19 \( 1 + (-0.253 - 0.0215i)T + (0.985 + 0.169i)T^{2} \)
23 \( 1 + (-0.778 + 0.628i)T^{2} \)
29 \( 1 + (0.372 - 0.927i)T^{2} \)
31 \( 1 + (1.03 - 1.67i)T + (-0.450 - 0.892i)T^{2} \)
37 \( 1 + (-0.915 + 0.511i)T + (0.524 - 0.851i)T^{2} \)
41 \( 1 + (-0.127 - 0.991i)T^{2} \)
43 \( 1 + (1.55 + 1.25i)T + (0.210 + 0.977i)T^{2} \)
47 \( 1 + (0.985 - 0.169i)T^{2} \)
53 \( 1 + (0.0424 - 0.999i)T^{2} \)
59 \( 1 + (-0.660 - 0.750i)T^{2} \)
61 \( 1 + (-1.98 - 0.253i)T + (0.967 + 0.251i)T^{2} \)
67 \( 1 + (0.787 - 0.241i)T + (0.828 - 0.559i)T^{2} \)
71 \( 1 + (-0.660 - 0.750i)T^{2} \)
73 \( 1 + (-0.577 + 0.0989i)T + (0.942 - 0.333i)T^{2} \)
79 \( 1 + (-0.943 - 0.830i)T + (0.127 + 0.991i)T^{2} \)
83 \( 1 + (0.210 - 0.977i)T^{2} \)
89 \( 1 + (0.778 + 0.628i)T^{2} \)
97 \( 1 + (0.500 - 1.41i)T + (-0.778 - 0.628i)T^{2} \)
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

−9.329621535153568548643965825807, −8.697801288234759937403233366555, −8.208831568625104543681526667478, −7.21614478044973310055477031147, −6.56292252948351922599776600777, −5.50838184930324924616110548509, −4.86749527010458469557855755277, −3.60629287181947314014805301242, −2.74784775290966912009826007522, −2.02877067151179075623063922580, 0.797865825233276197201233896893, 1.93004213577464842201279089640, 3.25283933264056583045702703924, 4.34360613809183591099523394192, 4.99107215360237761681707524613, 5.98704280384240828914323530948, 6.88258534577110833393567243654, 7.34530994900051699417757444245, 8.186472428316039784482469594405, 9.528434081158258405654366107125

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