Properties

Degree 2
Conductor $ 2 \cdot 5^{2} \cdot 7^{2} $
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 i·3-s − 4-s − 6-s + i·8-s + 2·9-s + 3·11-s + i·12-s − 2i·13-s + 16-s + 3i·17-s − 2i·18-s − 7·19-s − 3i·22-s + 24-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.353i·8-s + 0.666·9-s + 0.904·11-s + 0.288i·12-s − 0.554i·13-s + 0.250·16-s + 0.727i·17-s − 0.471i·18-s − 1.60·19-s − 0.639i·22-s + 0.204·24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.447 + 0.894i$
motivic weight  =  \(1\)
character  :  $\chi_{2450} (99, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2450,\ (\ :1/2),\ -0.447 + 0.894i)\)
\(L(1)\)  \(\approx\)  \(1.816632797\)
\(L(\frac12)\)  \(\approx\)  \(1.816632797\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + iT - 3T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 + 7T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 - 7iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 5iT - 73T^{2} \)
79 \( 1 + 14T + 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + 15T + 89T^{2} \)
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

−8.490532124872863079464037018826, −8.251577106685596022461795390188, −7.04842833833590880277776721134, −6.48129634868578868636556016924, −5.60306702001252148003430190571, −4.33478590734276780168099009521, −3.95806924599184681134781725135, −2.63760473033918254025977603676, −1.76616264277131175403557650458, −0.73743426868524249296873055799, 1.13931559641777263878283601826, 2.58014810226331224510076452612, 3.98041734576428427686310747813, 4.34178168622102925364911426728, 5.16460923396670724988183339617, 6.35922549668731555721178004746, 6.70162105190977187375468416688, 7.58151874902949358064537688703, 8.566767828534576884876706659597, 9.060403820093175901978861093502

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