Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.715 + 0.698i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (2.54 + 0.717i)7-s + 0.999·8-s + (−5.09 + 2.94i)11-s − 4.05·13-s + (−1.89 + 1.84i)14-s + (−0.5 + 0.866i)16-s + (0.371 − 0.214i)17-s + (−5.30 − 3.06i)19-s − 5.88i·22-s + (−0.876 + 1.51i)23-s + (2.02 − 3.51i)26-s + (−0.651 − 2.56i)28-s − 0.0419i·29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.962 + 0.271i)7-s + 0.353·8-s + (−1.53 + 0.886i)11-s − 1.12·13-s + (−0.506 + 0.493i)14-s + (−0.125 + 0.216i)16-s + (0.0901 − 0.0520i)17-s + (−1.21 − 0.703i)19-s − 1.25i·22-s + (−0.182 + 0.316i)23-s + (0.397 − 0.689i)26-s + (−0.123 − 0.484i)28-s − 0.00778i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.715 + 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.715 + 0.698i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.715 + 0.698i)$
$L(1)$  $\approx$  $0.8279025759$
$L(\frac12)$  $\approx$  $0.8279025759$
$L(\frac{3}{2})$   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 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.54 - 0.717i)T \)
good11 \( 1 + (5.09 - 2.94i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 4.05T + 13T^{2} \)
17 \( 1 + (-0.371 + 0.214i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.30 + 3.06i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.876 - 1.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.0419iT - 29T^{2} \)
31 \( 1 + (-7.92 + 4.57i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.928 - 0.536i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.61T + 41T^{2} \)
43 \( 1 + 11.0iT - 43T^{2} \)
47 \( 1 + (0.834 + 0.481i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.57 + 11.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.77 - 11.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.05 - 0.609i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-10.9 + 6.32i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.54iT - 71T^{2} \)
73 \( 1 + (4.66 + 8.08i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.35 + 9.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.1iT - 83T^{2} \)
89 \( 1 + (3.15 - 5.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.59T + 97T^{2} \)
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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

−8.406385102987290430929798440330, −7.77402866057765208488719557593, −7.35096464829947769213377544696, −6.42399365886341098691058878935, −5.41645057336213603542098899488, −4.88723181406656242715403005058, −4.27504565209181048809646460249, −2.55590249909829507351157577751, −2.05974087516458001665348294545, −0.32521830712250687671305067379, 1.00517070219244570131359075500, 2.30381143032202688272794875580, 2.84929315124293915180127352229, 4.13080810352585229543704563679, 4.80780645714587195592627813181, 5.56662470707408210884707960820, 6.58520103967957766471092421815, 7.68139197453259974306485983332, 8.104595643104299269007229362124, 8.553403154971262000408171068706

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