Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.313 - 0.949i$
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 + (−1.22 − 2.34i)7-s − 0.999·8-s + (2.03 − 1.17i)11-s − 4.64·13-s + (−2.64 − 0.107i)14-s + (−0.5 + 0.866i)16-s + (−3.95 + 2.28i)17-s + (−0.491 − 0.283i)19-s − 2.35i·22-s + (−2.91 + 5.04i)23-s + (−2.32 + 4.02i)26-s + (−1.41 + 2.23i)28-s − 2.55i·29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.464 − 0.885i)7-s − 0.353·8-s + (0.615 − 0.355i)11-s − 1.28·13-s + (−0.706 − 0.0287i)14-s + (−0.125 + 0.216i)16-s + (−0.958 + 0.553i)17-s + (−0.112 − 0.0650i)19-s − 0.502i·22-s + (−0.607 + 1.05i)23-s + (−0.455 + 0.789i)26-s + (−0.267 + 0.422i)28-s − 0.474i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.313 - 0.949i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.313 - 0.949i)$
$L(1)$  $\approx$  $0.5051997335$
$L(\frac12)$  $\approx$  $0.5051997335$
$L(\frac{3}{2})$   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 \( 1 + (1.22 + 2.34i)T \)
good11 \( 1 + (-2.03 + 1.17i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.64T + 13T^{2} \)
17 \( 1 + (3.95 - 2.28i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.491 + 0.283i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.91 - 5.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 2.55iT - 29T^{2} \)
31 \( 1 + (1.89 - 1.09i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-8.02 - 4.63i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.68T + 41T^{2} \)
43 \( 1 - 6.57iT - 43T^{2} \)
47 \( 1 + (5.46 + 3.15i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.07 - 10.5i)T + (-26.5 + 45.8i)T^{2} \)
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 \( 1 + (3.46 - 2.00i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.02iT - 71T^{2} \)
73 \( 1 + (4.10 + 7.11i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.13 - 7.15i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.171iT - 83T^{2} \)
89 \( 1 + (-2.72 + 4.72i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−9.143074498016195553573450155760, −8.001302763779181523183909951982, −7.34336761827572410710711679406, −6.44520859534432538260934316987, −5.86095410564953511135294527185, −4.64337963866448635372342890783, −4.18787502650409307270037387534, −3.28365098332425518973542533877, −2.35175973435294311944847237731, −1.19960443483372343684536820597, 0.13969305841174766703235277176, 2.16247421235809179573033221691, 2.79302193468133501328071604162, 4.05804446097860568465324244857, 4.68925017592901592463778050040, 5.55010418984547394506777399405, 6.30219634053850443789070094343, 6.94831460720384928917652667870, 7.62699223573744576660741916564, 8.561529267604898164501057750559

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