Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.785 - 0.619i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 - 0.619i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.785 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.785 - 0.619i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.785 - 0.619i)$
$L(1)$  $\approx$  $2.079739868$
$L(\frac12)$  $\approx$  $2.079739868$
$L(\frac{3}{2})$   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

−8.872937409779021016430858070192, −8.165553184461090283195398711803, −7.51325287422393432449838459466, −6.62135378403975435225893160954, −5.86870875473850998336554108552, −5.34275487936234806460470186615, −4.39491039716859746153379506833, −3.60262489016453675912230469579, −2.55087549470375620558136877583, −1.43025675900304902077840351559, 0.59691913888925619166841383613, 1.58320160684783934498302560862, 2.77287055808443594032770388554, 3.64646878012476560706104955270, 4.43515570542104617971403501796, 5.05734490718451188265690281104, 6.13776013367666741247343727391, 6.76930404344728578706607065062, 7.72656894843803651583937849687, 8.367707546697502838835821397003

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