Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.786 - 0.617i$
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 + (−0.358 + 2.62i)7-s + 0.999·8-s + (2.59 + 1.5i)11-s + 2.44·13-s + (2.44 − i)14-s + (−0.5 − 0.866i)16-s + (5.12 + 2.95i)17-s + (5.12 − 2.95i)19-s − 3i·22-s + (2.12 + 3.67i)23-s + (−1.22 − 2.12i)26-s + (−2.09 − 1.62i)28-s + 7.24i·29-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.135 + 0.990i)7-s + 0.353·8-s + (0.783 + 0.452i)11-s + 0.679·13-s + (0.654 − 0.267i)14-s + (−0.125 − 0.216i)16-s + (1.24 + 0.717i)17-s + (1.17 − 0.678i)19-s − 0.639i·22-s + (0.442 + 0.766i)23-s + (−0.240 − 0.416i)26-s + (−0.395 − 0.306i)28-s + 1.34i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.786 - 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.786 - 0.617i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.786 - 0.617i)$
$L(1)$  $\approx$  $1.594699939$
$L(\frac12)$  $\approx$  $1.594699939$
$L(\frac{3}{2})$   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 \( 1 + (0.358 - 2.62i)T \)
good11 \( 1 + (-2.59 - 1.5i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 + (-5.12 - 2.95i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.12 + 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.12 - 3.67i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.24iT - 29T^{2} \)
31 \( 1 + (7.86 + 4.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.210 - 0.121i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 11.8T + 41T^{2} \)
43 \( 1 - 0.242iT - 43T^{2} \)
47 \( 1 + (5.12 - 2.95i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.62 + 6.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.03 + 6.98i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.878 + 0.507i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-8.66 - 5i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.75iT - 71T^{2} \)
73 \( 1 + (0.717 - 1.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.37 + 2.38i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.63iT - 83T^{2} \)
89 \( 1 + (-5.19 - 9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−8.880259565034277725869139049790, −8.238427358564893244682242801841, −7.33962806294648404828811075265, −6.59880472717711817146172003116, −5.54180710896939671159707155779, −5.05452229837812848528705808571, −3.59550919960554892081953744550, −3.33569688737505150169847251417, −1.99040342845526136539756612032, −1.19987309093797477791013480713, 0.66437495239029434762558683254, 1.51264886013267997542706791999, 3.27105060440942964533861536903, 3.78629653214288560132585356280, 4.89880932891159006684498143449, 5.67912538851449596976445069719, 6.45994464125269434627862897246, 7.18636371421282152329497525934, 7.74694970549820460781591216609, 8.550284423652313402804260649223

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