Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.952 + 0.304i$
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 + (−0.717 − 2.54i)7-s + 0.999i·8-s + (−5.09 − 2.94i)11-s − 4.05i·13-s + (1.89 + 1.84i)14-s + (−0.5 − 0.866i)16-s + (0.214 − 0.371i)17-s + (5.30 − 3.06i)19-s + 5.88·22-s + (1.51 − 0.876i)23-s + (2.02 + 3.51i)26-s + (−2.56 − 0.651i)28-s − 0.0419i·29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.271 − 0.962i)7-s + 0.353i·8-s + (−1.53 − 0.886i)11-s − 1.12i·13-s + (0.506 + 0.493i)14-s + (−0.125 − 0.216i)16-s + (0.0520 − 0.0901i)17-s + (1.21 − 0.703i)19-s + 1.25·22-s + (0.316 − 0.182i)23-s + (0.397 + 0.689i)26-s + (−0.484 − 0.123i)28-s − 0.00778i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.952 + 0.304i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.952 + 0.304i)$
$L(1)$  $\approx$  $0.5556914269$
$L(\frac12)$  $\approx$  $0.5556914269$
$L(\frac{3}{2})$   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 \( 1 + (0.717 + 2.54i)T \)
good11 \( 1 + (5.09 + 2.94i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 4.05iT - 13T^{2} \)
17 \( 1 + (-0.214 + 0.371i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.30 + 3.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.51 + 0.876i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.0419iT - 29T^{2} \)
31 \( 1 + (-7.92 - 4.57i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.536 + 0.928i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.61T + 41T^{2} \)
43 \( 1 + 11.0T + 43T^{2} \)
47 \( 1 + (-0.481 - 0.834i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (11.3 + 6.57i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (6.77 - 11.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.05 + 0.609i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.32 + 10.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.54iT - 71T^{2} \)
73 \( 1 + (8.08 + 4.66i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.35 + 9.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 + (-3.15 - 5.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 2.59iT - 97T^{2} \)
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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

−8.061134624234165319928146881363, −7.81917372972862256226378283057, −7.00733491271572408488388842284, −6.17111484421293586620998867283, −5.31829649353847434744930221407, −4.72850814229054200659494173547, −3.22328166390438890052194284972, −2.83978269790202113571703184537, −1.10489284966606488781944755219, −0.23542657453442302119216379697, 1.53928974421368396235402789064, 2.46527055969234939250478265822, 3.13101048266434104906384408383, 4.39200868998066747645900813178, 5.19296842233016523135650871185, 6.03573935644101415658125296777, 6.91924089525061338691099212391, 7.72932031173410658148658055099, 8.246467263798671124286875154766, 9.135618635886515353624666773906

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