Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.349 - 0.936i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (2.63 − 0.189i)7-s − 0.999·8-s + (3.44 + 1.99i)11-s + 0.0681·13-s + (1.48 + 2.19i)14-s + (−0.5 − 0.866i)16-s + (6.34 + 3.66i)17-s + (1.76 − 1.01i)19-s + 3.98i·22-s + (−1.86 − 3.23i)23-s + (0.0340 + 0.0590i)26-s + (−1.15 + 2.38i)28-s − 0.898i·29-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.997 − 0.0716i)7-s − 0.353·8-s + (1.03 + 0.600i)11-s + 0.0189·13-s + (0.396 + 0.585i)14-s + (−0.125 − 0.216i)16-s + (1.53 + 0.888i)17-s + (0.404 − 0.233i)19-s + 0.848i·22-s + (−0.389 − 0.673i)23-s + (0.00668 + 0.0115i)26-s + (−0.218 + 0.449i)28-s − 0.166i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.349 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.349 - 0.936i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.349 - 0.936i)$
$L(1)$  $\approx$  $2.801846888$
$L(\frac12)$  $\approx$  $2.801846888$
$L(\frac{3}{2})$   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 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.63 + 0.189i)T \)
good11 \( 1 + (-3.44 - 1.99i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 0.0681T + 13T^{2} \)
17 \( 1 + (-6.34 - 3.66i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.76 + 1.01i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.86 + 3.23i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.898iT - 29T^{2} \)
31 \( 1 + (4.18 + 2.41i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.52 - 2.03i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.68T + 41T^{2} \)
43 \( 1 - 0.964iT - 43T^{2} \)
47 \( 1 + (-1.43 + 0.830i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.61 + 11.4i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.32 + 9.21i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.51 + 3.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.23 + 5.33i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.93iT - 71T^{2} \)
73 \( 1 + (5.82 - 10.0i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.77 - 15.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 14.3iT - 83T^{2} \)
89 \( 1 + (-0.913 - 1.58i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 17.1T + 97T^{2} \)
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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

−8.483928960096666577100941007817, −8.129117303197815773568401161319, −7.25464578277839164779260377080, −6.65978938754613407695935553205, −5.67125061011558069627318242177, −5.13252757496891263322980689224, −4.12920576100040620023324277408, −3.63017453870355683297213468065, −2.20338529943480945196063638614, −1.14991096354384808858007785131, 0.973948424704615278154070501327, 1.73106481391663652186129747905, 3.00798163197065683192960654202, 3.71928431960114171338843909693, 4.58480098561863554919916505543, 5.51990774945949546214482032566, 5.88959503723011581787115443755, 7.22719956503191091396495194739, 7.68944890909445013139817218892, 8.803226346972639176435422430965

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