Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.876 - 0.481i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 + 0.707i)7-s + (−0.707 − 0.707i)8-s − 3.41i·11-s + (−2.02 + 2.02i)13-s + 1.00·14-s − 1.00·16-s + (−4.37 + 4.37i)17-s − 4.87i·19-s + (−2.41 − 2.41i)22-s + (−3.74 − 3.74i)23-s + 2.86i·26-s + (0.707 − 0.707i)28-s − 5.21·29-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (0.267 + 0.267i)7-s + (−0.250 − 0.250i)8-s − 1.02i·11-s + (−0.561 + 0.561i)13-s + 0.267·14-s − 0.250·16-s + (−1.06 + 1.06i)17-s − 1.11i·19-s + (−0.514 − 0.514i)22-s + (−0.780 − 0.780i)23-s + 0.561i·26-s + (0.133 − 0.133i)28-s − 0.968·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.876 - 0.481i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.876 - 0.481i)$
$L(1)$  $\approx$  $0.4060791145$
$L(\frac12)$  $\approx$  $0.4060791145$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.707 - 0.707i)T \)
good11 \( 1 + 3.41iT - 11T^{2} \)
13 \( 1 + (2.02 - 2.02i)T - 13iT^{2} \)
17 \( 1 + (4.37 - 4.37i)T - 17iT^{2} \)
19 \( 1 + 4.87iT - 19T^{2} \)
23 \( 1 + (3.74 + 3.74i)T + 23iT^{2} \)
29 \( 1 + 5.21T + 29T^{2} \)
31 \( 1 + 4.93T + 31T^{2} \)
37 \( 1 + (-6.87 - 6.87i)T + 37iT^{2} \)
41 \( 1 - 8.77iT - 41T^{2} \)
43 \( 1 + (-0.174 + 0.174i)T - 43iT^{2} \)
47 \( 1 + (-3.42 + 3.42i)T - 47iT^{2} \)
53 \( 1 + (6.43 + 6.43i)T + 53iT^{2} \)
59 \( 1 + 2.22T + 59T^{2} \)
61 \( 1 + 15.2T + 61T^{2} \)
67 \( 1 + (3.81 + 3.81i)T + 67iT^{2} \)
71 \( 1 + 10.9iT - 71T^{2} \)
73 \( 1 + (-2.51 + 2.51i)T - 73iT^{2} \)
79 \( 1 + 9.98iT - 79T^{2} \)
83 \( 1 + (2.35 + 2.35i)T + 83iT^{2} \)
89 \( 1 + 4.07T + 89T^{2} \)
97 \( 1 + (-5.64 - 5.64i)T + 97iT^{2} \)
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

−8.374593062732835602789702398551, −7.52590068478585856590374301353, −6.37221602567244998447168848948, −6.11819950997176178787267205660, −4.92489932920894712491710903124, −4.41479287538218029093758055017, −3.43368424519402483816116583796, −2.49896099946250439864773525395, −1.65167427656059872788116592166, −0.094847808979907444558789489842, 1.77421653523002986615034036355, 2.69069404878822044393461544350, 3.91266786046779354499690526490, 4.43098951412949537428018522903, 5.39770132957090487096312581966, 5.91652764237828878123399137645, 7.12479213755720623315989995198, 7.41582712146419793937548354774, 8.070091890593712631223575224369, 9.231638711087065080546929268853

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