Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.894 - 0.447i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s i·7-s + i·8-s i·13-s − 14-s + 16-s − 3i·17-s − 2·19-s + 3i·23-s − 26-s + i·28-s − 3·29-s − 31-s i·32-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.377i·7-s + 0.353i·8-s − 0.277i·13-s − 0.267·14-s + 0.250·16-s − 0.727i·17-s − 0.458·19-s + 0.625i·23-s − 0.196·26-s + 0.188i·28-s − 0.557·29-s − 0.179·31-s − 0.176i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.894 - 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.894 - 0.447i)$
$L(1)$  $\approx$  $0.5489423164$
$L(\frac12)$  $\approx$  $0.5489423164$
$L(\frac{3}{2})$   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 \( 1 + iT \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + iT \)
good11 \( 1 + 11T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + 3T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 15iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−8.365065907294969951319741632313, −7.53796247359455911496786008721, −6.88069824419769952242560770924, −5.79772447578655773769943597364, −5.09722438654297425079593523972, −4.17885976969187413867762469923, −3.44089148561824895520842404339, −2.49738694601324945227769951780, −1.45381212515555590144940713589, −0.17003627585707894682212660472, 1.50070957336960335228174065644, 2.68174709079835084957881115169, 3.79512642451380361793118861554, 4.56497589414483073582353861239, 5.40919411310355706303595223420, 6.20666347786653201093634544482, 6.73372141171708547299939532131, 7.67054796206946527669475760269, 8.332642460642684211519171969751, 8.926240425076139529861135002547

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