Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.433 - 0.901i$
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 + (−2.34 − 1.22i)7-s − 0.999i·8-s + (−2.03 − 1.17i)11-s + 4.64i·13-s + (−2.64 + 0.107i)14-s + (−0.5 − 0.866i)16-s + (−2.28 + 3.95i)17-s + (0.491 − 0.283i)19-s − 2.35·22-s + (5.04 − 2.91i)23-s + (2.32 + 4.02i)26-s + (−2.23 + 1.41i)28-s + 2.55i·29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.885 − 0.464i)7-s − 0.353i·8-s + (−0.615 − 0.355i)11-s + 1.28i·13-s + (−0.706 + 0.0287i)14-s + (−0.125 − 0.216i)16-s + (−0.553 + 0.958i)17-s + (0.112 − 0.0650i)19-s − 0.502·22-s + (1.05 − 0.607i)23-s + (0.455 + 0.789i)26-s + (−0.422 + 0.267i)28-s + 0.474i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.433 - 0.901i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.433 - 0.901i)$
$L(1)$  $\approx$  $1.385209669$
$L(\frac12)$  $\approx$  $1.385209669$
$L(\frac{3}{2})$   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 \( 1 + (2.34 + 1.22i)T \)
good11 \( 1 + (2.03 + 1.17i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.64iT - 13T^{2} \)
17 \( 1 + (2.28 - 3.95i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.491 + 0.283i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.04 + 2.91i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 2.55iT - 29T^{2} \)
31 \( 1 + (1.89 + 1.09i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.63 - 8.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.68T + 41T^{2} \)
43 \( 1 + 6.57T + 43T^{2} \)
47 \( 1 + (-3.15 - 5.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-10.5 - 6.07i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.67 - 2.90i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.85 - 3.95i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.00 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.02iT - 71T^{2} \)
73 \( 1 + (-7.11 - 4.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.13 - 7.15i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 0.171T + 83T^{2} \)
89 \( 1 + (-2.72 - 4.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 10.8iT - 97T^{2} \)
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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

−8.936921760481493462039809644610, −8.103679575565471546696899391164, −6.93595676392368715623652253596, −6.64427921784762187786783567131, −5.78336776851344744475408557857, −4.80337589758956622749306225994, −4.11088417834676815495736752904, −3.27383281292110736983139418071, −2.43858155800744158202956994845, −1.23073506466915327065917580952, 0.34824325723573838997817622041, 2.22815964022295487314061009895, 3.02700440910445841536465449834, 3.69624027699207786119685211172, 5.01327233025602043736626325708, 5.34051815024111638002127753820, 6.22370094108809353614437665548, 7.05065493395910546178552190735, 7.58312466329767653537291953435, 8.494426157767703898564192606065

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