Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.995 + 0.0996i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−2.43 + 1.04i)7-s − 0.999i·8-s + (−1.38 − 0.800i)11-s − 0.770i·13-s + (−1.58 + 2.11i)14-s + (−0.5 − 0.866i)16-s + (1.76 − 3.05i)17-s + (3.06 − 1.77i)19-s − 1.60·22-s + (−2.79 + 1.61i)23-s + (−0.385 − 0.667i)26-s + (−0.313 + 2.62i)28-s + 0.700i·29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.919 + 0.393i)7-s − 0.353i·8-s + (−0.417 − 0.241i)11-s − 0.213i·13-s + (−0.423 + 0.566i)14-s + (−0.125 − 0.216i)16-s + (0.427 − 0.739i)17-s + (0.703 − 0.406i)19-s − 0.341·22-s + (−0.582 + 0.336i)23-s + (−0.0755 − 0.130i)26-s + (−0.0592 + 0.496i)28-s + 0.130i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.995 + 0.0996i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.995 + 0.0996i)$
$L(1)$  $\approx$  $0.8428912823$
$L(\frac12)$  $\approx$  $0.8428912823$
$L(\frac{3}{2})$   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 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.43 - 1.04i)T \)
good11 \( 1 + (1.38 + 0.800i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.770iT - 13T^{2} \)
17 \( 1 + (-1.76 + 3.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.06 + 1.77i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.79 - 1.61i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 0.700iT - 29T^{2} \)
31 \( 1 + (-1.13 - 0.656i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.457 + 0.792i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 4.88T + 41T^{2} \)
43 \( 1 + 9.26T + 43T^{2} \)
47 \( 1 + (1.33 + 2.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (8.04 + 4.64i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.56 - 2.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.43 - 5.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.40 - 5.90i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 6.47iT - 71T^{2} \)
73 \( 1 + (9.55 + 5.51i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.45 + 2.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.9T + 83T^{2} \)
89 \( 1 + (4.40 + 7.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 5.31iT - 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−8.361379120701471273418354034676, −7.41837994242045012362610220613, −6.73343103659213017779869688141, −5.85447137923868990927642096421, −5.31102780709569474410273689891, −4.43316752892547827627852843132, −3.18264149899749915546824838617, −3.01850515311275791169592060604, −1.67870094024664969655678537644, −0.19658346833989801788308799134, 1.57337527119930951647873223560, 2.86693774490867143342641215848, 3.55886870016635647765662158298, 4.37906518646794948956644545353, 5.26036189705562371646215099722, 6.12251206422401749457851126254, 6.61490877384186752225029292490, 7.52518421798985401874154284961, 8.057895815510881535448991841240, 8.979130238430015981816894903063

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