Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.231 + 0.972i$
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 + (−0.189 − 2.63i)7-s + 0.999i·8-s + (2.55 + 1.47i)11-s − 3.93i·13-s + (1.48 + 2.19i)14-s + (−0.5 − 0.866i)16-s + (0.199 − 0.346i)17-s + (0.0305 − 0.0176i)19-s − 2.94·22-s + (−3.23 + 1.86i)23-s + (1.96 + 3.40i)26-s + (−2.38 − 1.15i)28-s − 8.89i·29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.0716 − 0.997i)7-s + 0.353i·8-s + (0.769 + 0.444i)11-s − 1.09i·13-s + (0.396 + 0.585i)14-s + (−0.125 − 0.216i)16-s + (0.0484 − 0.0839i)17-s + (0.00700 − 0.00404i)19-s − 0.628·22-s + (−0.673 + 0.389i)23-s + (0.385 + 0.667i)26-s + (−0.449 − 0.218i)28-s − 1.65i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.231 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.231 + 0.972i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.231 + 0.972i)$
$L(1)$  $\approx$  $0.9604714722$
$L(\frac12)$  $\approx$  $0.9604714722$
$L(\frac{3}{2})$   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 \( 1 + (0.189 + 2.63i)T \)
good11 \( 1 + (-2.55 - 1.47i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 3.93iT - 13T^{2} \)
17 \( 1 + (-0.199 + 0.346i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.0305 + 0.0176i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.23 - 1.86i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.89iT - 29T^{2} \)
31 \( 1 + (-0.717 - 0.414i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.96 + 6.86i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.31T + 41T^{2} \)
43 \( 1 - 3.03T + 43T^{2} \)
47 \( 1 + (-2.90 - 5.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.72 - 2.14i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.78 + 4.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.97 - 5.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.25 - 10.8i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.93iT - 71T^{2} \)
73 \( 1 + (-0.297 - 0.171i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.15 + 7.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.3T + 83T^{2} \)
89 \( 1 + (3.08 + 5.34i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 15.6iT - 97T^{2} \)
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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

−8.403096654444892770476015731747, −7.47094854580024458151779143394, −7.37300210489151553178339873056, −6.20240419590698298210812274915, −5.71069829243081729408260651299, −4.48885688740841652859696381316, −3.86118122374679568593480465307, −2.67824660660693284817720585860, −1.44616642469027448073438007462, −0.38830614694181141914515757695, 1.31659106807686206814217734119, 2.20894627774537117853757703480, 3.18715458234126220354981940192, 4.06810800868188482742715656300, 5.05638529689564962963254232305, 6.08118481757207300746466329137, 6.62916956839217891096604533340, 7.47265838682520022577210721126, 8.520431101661453294749734185961, 8.849785654079806902784204198575

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