Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Sign $0.921 + 0.387i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + (−1.59 − 2.76i)7-s + 9-s + (2.11 + 3.66i)11-s + (1.09 − 1.88i)13-s − 15-s + (−2.56 + 4.43i)17-s + (−0.325 + 0.564i)19-s + (1.59 + 2.76i)21-s + (4.02 − 6.96i)23-s + 25-s − 27-s + (−3.94 − 6.83i)29-s + (4.23 + 7.34i)31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + (−0.602 − 1.04i)7-s + 0.333·9-s + (0.638 + 1.10i)11-s + (0.302 − 0.523i)13-s − 0.258·15-s + (−0.621 + 1.07i)17-s + (−0.0747 + 0.129i)19-s + (0.347 + 0.602i)21-s + (0.838 − 1.45i)23-s + 0.200·25-s − 0.192·27-s + (−0.733 − 1.27i)29-s + (0.761 + 1.31i)31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4020\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 67\)
\( \varepsilon \)  =  $0.921 + 0.387i$
motivic weight  =  \(1\)
character  :  $\chi_{4020} (3781, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4020,\ (\ :1/2),\ 0.921 + 0.387i)$
$L(1)$  $\approx$  $1.562949375$
$L(\frac12)$  $\approx$  $1.562949375$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;5,\;67\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;67\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
67 \( 1 + (-5.59 - 5.97i)T \)
good7 \( 1 + (1.59 + 2.76i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.11 - 3.66i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.09 + 1.88i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.56 - 4.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.325 - 0.564i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.02 + 6.96i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.94 + 6.83i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.23 - 7.34i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.75 - 3.03i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.135 - 0.234i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 8.32T + 43T^{2} \)
47 \( 1 + (0.171 + 0.297i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 7.89T + 53T^{2} \)
59 \( 1 + 4.39T + 59T^{2} \)
61 \( 1 + (7.05 - 12.2i)T + (-30.5 - 52.8i)T^{2} \)
71 \( 1 + (-1.45 - 2.51i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.18 + 5.51i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.598 - 1.03i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.963 + 1.66i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 3.23T + 89T^{2} \)
97 \( 1 + (-5.58 + 9.67i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.448993553663534082547952844653, −7.45509242838139544429537272385, −6.75716799671788973692770937819, −6.38207965980780403112369416568, −5.51550609275166861628065317202, −4.38526242637266340904440570214, −4.12915228874347625224774600124, −2.89313945188613768049542772971, −1.73594354325011122156613726747, −0.70308317341682448871715088178, 0.798963982856257673985654999283, 2.03226374180278064634066167084, 3.01067192267472527827193574935, 3.82657633013203640532752956288, 4.97941165329859241402556608251, 5.62290629488595472155805944514, 6.20445648894534011015953716717, 6.81155515879638511274564564770, 7.65278292555379184987498264851, 8.834918720449767846210195869349

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