Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.738 + 0.673i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + (−0.648 − 2.56i)7-s − 8-s + 1.29i·11-s + 3.13·13-s + (0.648 + 2.56i)14-s + 16-s − 5.53i·17-s − 7.37i·19-s − 1.29i·22-s − 1.83·23-s − 3.13·26-s + (−0.648 − 2.56i)28-s + 1.83i·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.245 − 0.969i)7-s − 0.353·8-s + 0.390i·11-s + 0.868·13-s + (0.173 + 0.685i)14-s + 0.250·16-s − 1.34i·17-s − 1.69i·19-s − 0.276i·22-s − 0.382·23-s − 0.613·26-s + (−0.122 − 0.484i)28-s + 0.340i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.738 + 0.673i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (3149, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.738 + 0.673i)$
$L(1)$  $\approx$  $0.7906938614$
$L(\frac12)$  $\approx$  $0.7906938614$
$L(\frac{3}{2})$   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 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (0.648 + 2.56i)T \)
good11 \( 1 - 1.29iT - 11T^{2} \)
13 \( 1 - 3.13T + 13T^{2} \)
17 \( 1 + 5.53iT - 17T^{2} \)
19 \( 1 + 7.37iT - 19T^{2} \)
23 \( 1 + 1.83T + 23T^{2} \)
29 \( 1 - 1.83iT - 29T^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 + 10.6iT - 37T^{2} \)
41 \( 1 + 3.13T + 41T^{2} \)
43 \( 1 + 3.53iT - 43T^{2} \)
47 \( 1 - 10.7iT - 47T^{2} \)
53 \( 1 + 4.42T + 53T^{2} \)
59 \( 1 - 7.18T + 59T^{2} \)
61 \( 1 + 4.88iT - 61T^{2} \)
67 \( 1 + 9.79iT - 67T^{2} \)
71 \( 1 - 7.37iT - 71T^{2} \)
73 \( 1 - 3.40T + 73T^{2} \)
79 \( 1 + 9.01T + 79T^{2} \)
83 \( 1 + 6.26iT - 83T^{2} \)
89 \( 1 + 7.94T + 89T^{2} \)
97 \( 1 - 8.09T + 97T^{2} \)
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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

−8.510277468724149706717962251755, −7.52150131432518883516318002252, −7.02879613815959884026012466239, −6.49020694044322157494031778846, −5.31185302187272300161989001208, −4.52237221878034315693262732311, −3.49298328039080426350897515640, −2.64788163180643589833755482943, −1.36398880288701368453831445973, −0.32716838526205343209583852164, 1.36898029819290793600333648359, 2.23229198507400689323762368072, 3.37580617732920618778837566773, 4.07549201647680555938071982613, 5.53067003765113477509374586583, 6.04177841606777977786827021806, 6.54543053835256418932611118600, 7.83867453178346897103708521442, 8.329754234276923904592652083888, 8.726187833825493114843509671044

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