Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.0239 - 0.999i$
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 + (−1.04 − 2.43i)7-s + 0.999·8-s + (1.38 + 0.800i)11-s − 0.770·13-s + (−1.58 + 2.11i)14-s + (−0.5 − 0.866i)16-s + (−3.05 − 1.76i)17-s + (−3.06 + 1.77i)19-s − 1.60i·22-s + (−1.61 − 2.79i)23-s + (0.385 + 0.667i)26-s + (2.62 + 0.313i)28-s + 0.700i·29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.393 − 0.919i)7-s + 0.353·8-s + (0.417 + 0.241i)11-s − 0.213·13-s + (−0.423 + 0.566i)14-s + (−0.125 − 0.216i)16-s + (−0.739 − 0.427i)17-s + (−0.703 + 0.406i)19-s − 0.341i·22-s + (−0.336 − 0.582i)23-s + (0.0755 + 0.130i)26-s + (0.496 + 0.0592i)28-s + 0.130i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0239 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0239 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.0239 - 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.0239 - 0.999i)$
$L(1)$  $\approx$  $0.3179976258$
$L(\frac12)$  $\approx$  $0.3179976258$
$L(\frac{3}{2})$   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 \( 1 + (1.04 + 2.43i)T \)
good11 \( 1 + (-1.38 - 0.800i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 0.770T + 13T^{2} \)
17 \( 1 + (3.05 + 1.76i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.06 - 1.77i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.61 + 2.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.700iT - 29T^{2} \)
31 \( 1 + (-1.13 - 0.656i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.792 + 0.457i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 4.88T + 41T^{2} \)
43 \( 1 - 9.26iT - 43T^{2} \)
47 \( 1 + (2.31 - 1.33i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.64 + 8.04i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.56 - 2.70i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (9.43 - 5.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (5.90 + 3.40i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.47iT - 71T^{2} \)
73 \( 1 + (5.51 - 9.55i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.45 - 2.51i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 11.9iT - 83T^{2} \)
89 \( 1 + (4.40 + 7.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.31T + 97T^{2} \)
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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

−8.981160084065101118023399677391, −8.209781096169209263950376412124, −7.41392590119986509306596793998, −6.73812175803198460976215203135, −5.98967956249694605821624132378, −4.61115616362330712688478276241, −4.22234224159307388202088889143, −3.21932810283431984105816595790, −2.27584668673057590262649973914, −1.13770521350681452733081460032, 0.11861745090790760417584815255, 1.74203017571324695216495356635, 2.71371011496764991604001687883, 3.87897681745334184470543739781, 4.74368297513672643566361303371, 5.71396983544724979736130400146, 6.23003526969703848492226203326, 6.93244139806186851137376254469, 7.79827879491635583301954500151, 8.619148621534943977202913814713

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