Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + i·3-s − 4-s + 6-s i·7-s + i·8-s − 9-s − 4·11-s i·12-s + 2i·13-s − 14-s + 16-s − 6i·17-s + i·18-s + 21-s + 4i·22-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.408·6-s − 0.377i·7-s + 0.353i·8-s − 0.333·9-s − 1.20·11-s − 0.288i·12-s + 0.554i·13-s − 0.267·14-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s + 0.218·21-s + 0.852i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1050\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.894 - 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{1050} (799, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  1
Selberg data  =  $(2,\ 1050,\ (\ :1/2),\ -0.894 - 0.447i)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 - iT \)
5 \( 1 \)
7 \( 1 + iT \)
good11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 8iT - 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 4iT - 47T^{2} \)
53 \( 1 + 10iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 6iT - 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 4iT - 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
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

−9.525039483962604332555282034981, −8.998605946800609538118590352237, −7.75119734905749421249091080610, −7.17199259741541919247383076839, −5.56965946779049544365706938216, −5.07252470279251523848864609082, −3.92928183227287822437359444945, −3.10998675331236431894781811691, −1.88864307716317065101849509719, 0, 1.91466993522813114027894941489, 3.16112833952143115810703523149, 4.44883157320725270432133685311, 5.60609386748082025149710631782, 6.01192522266309615903561756089, 7.15479985705976186082730493055, 7.87554170069646117703827877949, 8.476667599480528572984716795808, 9.323845620263769474202194793591

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