Properties

Degree 2
Conductor $ 2 \cdot 5^{2} \cdot 7^{2} $
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 + 2i·3-s − 4-s − 2·6-s i·8-s − 9-s − 2i·12-s + 4i·13-s + 16-s + 6i·17-s i·18-s + 2·19-s + 2·24-s − 4·26-s + 4i·27-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.15i·3-s − 0.5·4-s − 0.816·6-s − 0.353i·8-s − 0.333·9-s − 0.577i·12-s + 1.10i·13-s + 0.250·16-s + 1.45i·17-s − 0.235i·18-s + 0.458·19-s + 0.408·24-s − 0.784·26-s + 0.769i·27-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.894 + 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{2450} (99, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 2450,\ (\ :1/2),\ -0.894 + 0.447i)$
$L(1)$  $\approx$  $1.420893954$
$L(\frac12)$  $\approx$  $1.420893954$
$L(\frac{3}{2})$   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 \( 1 - 2iT - 3T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 6iT - 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 - 8iT - 43T^{2} \)
47 \( 1 + 12iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 6T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 - 4iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 2iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 6iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
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

−9.291580408060141983979399186092, −8.686799213988998412180612357826, −7.974685124744191518425915416230, −6.94757629042622922564060045741, −6.30169340373003154799973259917, −5.41015405814042970595648062909, −4.52338079867262475696931665972, −4.08804702133336435501210645596, −3.10579981581082720866300495059, −1.54539346394132368876531348887, 0.51355078800606068609580219707, 1.39715208919112226567774463728, 2.60263928118040302973129514711, 3.17540836021014981116902664595, 4.52680662177885794476002978576, 5.29080240231530046836053673283, 6.26241248410960786189616124972, 7.07256928059767764542463318469, 7.77997128109805632868382916423, 8.372317926425118838681390711910

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