Properties

Degree 2
Conductor $ 2^{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·7-s + 6·11-s + 2i·13-s + 4·19-s + 6i·23-s + 6·29-s + 8·31-s − 2i·37-s − 12·41-s − 4i·43-s + 12i·47-s − 49-s + 6i·53-s − 10·61-s − 8i·67-s + ⋯
L(s)  = 1  − 0.377i·7-s + 1.80·11-s + 0.554i·13-s + 0.917·19-s + 1.25i·23-s + 1.11·29-s + 1.43·31-s − 0.328i·37-s − 1.87·41-s − 0.609i·43-s + 1.75i·47-s − 0.142·49-s + 0.824i·53-s − 1.28·61-s − 0.977i·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6300\)    =    \(2^{2} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.894 - 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{6300} (6049, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 6300,\ (\ :1/2),\ 0.894 - 0.447i)\)
\(L(1)\)  \(\approx\)  \(2.399372177\)
\(L(\frac12)\)  \(\approx\)  \(2.399372177\)
\(L(\frac{3}{2})\)   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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + iT \)
good11 \( 1 - 6T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 12iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 12iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−7.990084037998074328060509500241, −7.38827832652121158513470294345, −6.52017477764552950444850539455, −6.27549974579458084361128064107, −5.12358101024344324129832713009, −4.43040979552528500862834811718, −3.69508927659995564259377443052, −2.99850710010784147513581177145, −1.65603105868881597132470199974, −1.03261491511701919146318739714, 0.74438357632839803568015666113, 1.66048934383948113939919306771, 2.81514690297851522087089700088, 3.48434347903952138590673767717, 4.42735300891593009444764014884, 5.03256300300425476626843118864, 6.01516525757656425775262953203, 6.58274237140609938530377426043, 7.09202293056876309228527824723, 8.248811662233940481852217313048

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