Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.768 - 0.639i$
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 + (1.62 + 2.09i)7-s − 0.999i·8-s + (2.59 + 1.5i)11-s + 2.44i·13-s + (2.44 + 0.999i)14-s + (−0.5 − 0.866i)16-s + (−0.507 + 0.878i)17-s + (−0.878 + 0.507i)19-s + 3·22-s + (−3.67 + 2.12i)23-s + (1.22 + 2.12i)26-s + (2.62 − 0.358i)28-s + 1.24i·29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.612 + 0.790i)7-s − 0.353i·8-s + (0.783 + 0.452i)11-s + 0.679i·13-s + (0.654 + 0.267i)14-s + (−0.125 − 0.216i)16-s + (−0.123 + 0.213i)17-s + (−0.201 + 0.116i)19-s + 0.639·22-s + (−0.766 + 0.442i)23-s + (0.240 + 0.416i)26-s + (0.495 − 0.0677i)28-s + 0.230i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.768 - 0.639i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.768 - 0.639i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.768 - 0.639i)$
$L(1)$  $\approx$  $2.780289142$
$L(\frac12)$  $\approx$  $2.780289142$
$L(\frac{3}{2})$   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 \( 1 + (-1.62 - 2.09i)T \)
good11 \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 + (0.507 - 0.878i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.878 - 0.507i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.67 - 2.12i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 1.24iT - 29T^{2} \)
31 \( 1 + (-4.86 - 2.80i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.12 - 7.13i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 2.02T + 41T^{2} \)
43 \( 1 + 8.24T + 43T^{2} \)
47 \( 1 + (0.507 + 0.878i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.07 - 0.621i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.76 + 9.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.12 + 2.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5 - 8.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 + (7.24 + 4.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.62 - 9.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 3.16T + 83T^{2} \)
89 \( 1 + (5.19 + 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 3.76iT - 97T^{2} \)
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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

−8.677344708707020723745881790432, −8.201407846138360551342858366019, −7.03927178971431411878170296323, −6.43600298041568673007329371329, −5.64345102024019245175992096971, −4.77741234492252665186400252493, −4.19322323973446216009098073691, −3.19470665456598952431332159511, −2.11719480203679649183899570274, −1.43011688794251632095660502580, 0.70558779521651691287186195459, 2.02829338937267211144962754552, 3.18058444351203536781635439010, 4.05903620542300177525420504881, 4.61403630259292776657960119780, 5.59070189509363875684863529177, 6.30055599212651985442746816882, 7.03538933370464312075480138751, 7.83703031238422927227934126385, 8.347409702128952980881043253731

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