Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.999 - 0.0387i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.999 - 0.0387i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2843, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.999 - 0.0387i)$
$L(1)$  $\approx$  $0.9867969435$
$L(\frac12)$  $\approx$  $0.9867969435$
$L(\frac{3}{2})$   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

−8.958418637674591178201778976577, −8.229105983486275502354226311219, −7.55898453671508410147842335011, −6.49645036941425558441788707570, −6.19677680468077533436546821746, −5.30180247174351350408890308783, −4.45973671274920288878868079210, −3.56374265251577514429304408744, −2.84600117698028953473279023127, −1.54405428099446043313228903992, 0.23975981702927033148530612956, 1.74711884951506192461190227099, 2.52691134046953549203251126106, 3.67227151550542258534234427383, 4.28353486804820298965570651734, 5.08657332411245145289609557402, 6.01695858995007830514503795756, 6.74030019203316120967922891196, 7.39106027990140832355212970676, 8.527015355444187232356656556779

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