Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.425 + 0.904i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.425 + 0.904i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.425 + 0.904i)$
$L(1)$  $\approx$  $3.191263611$
$L(\frac12)$  $\approx$  $3.191263611$
$L(\frac{3}{2})$   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

−8.530393437099335892147866417621, −7.61899171658359570318123536281, −7.13699805960238067867587734834, −6.14649921052744335155745952407, −5.31680967606321012072246604910, −4.62536132333522771274203628959, −3.93046307187542501176072534093, −2.93862674397594143016708173191, −1.90311166195754268043352668512, −0.913238477610655555928773736100, 1.25616347407841914521346433618, 2.33997250473659104423183655890, 3.39227307848600555962783759695, 4.23509374622079660789663647894, 4.98992762787523237133191323629, 5.87498890704531483914833441581, 6.24819820078679528868883235870, 7.52885322528370833088040230292, 7.84707432000188627595708349076, 8.674095241707565205393889162797

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