Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.447 + 0.894i$
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 + 4·11-s + 2i·13-s − 14-s + 16-s − 2i·17-s − 4·19-s − 4i·22-s − 8i·23-s + 2·26-s + i·28-s − 2·29-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.377i·7-s + 0.353i·8-s + 1.20·11-s + 0.554i·13-s − 0.267·14-s + 0.250·16-s − 0.485i·17-s − 0.917·19-s − 0.852i·22-s − 1.66i·23-s + 0.392·26-s + 0.188i·28-s − 0.371·29-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.447 + 0.894i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.447 + 0.894i)$
$L(1)$  $\approx$  $1.589368584$
$L(\frac12)$  $\approx$  $1.589368584$
$L(\frac{3}{2})$   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 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 8iT - 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 6T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 12iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 - 2iT - 97T^{2} \)
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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

−8.647070953951551054909897349010, −7.85087508873386847634823185653, −6.67989243994877390682436296510, −6.45051981026895309423706895960, −5.13370250830456321850211520648, −4.31857800866127595740348614604, −3.79992080817952944918219692036, −2.66116301738036995538247288159, −1.73257711146949815551442522475, −0.56464863805919409306505491886, 1.13239738017760799271677146140, 2.35889104555269450690370290147, 3.71753193214688542609658064791, 4.13904590080917166872358906659, 5.45177830962152926500253079278, 5.78842418283782846665269507767, 6.73308660960340395096537949757, 7.33013367924308343034170547799, 8.212943581389159326109155280598, 8.826720435876726295317263037843

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