Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.683 - 0.729i$
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 + (−2.09 − 1.62i)7-s + 0.999·8-s + (−2.59 + 1.5i)11-s + 2.44·13-s + (2.44 − 0.999i)14-s + (−0.5 + 0.866i)16-s + (0.878 − 0.507i)17-s + (0.878 + 0.507i)19-s − 3i·22-s + (−2.12 + 3.67i)23-s + (−1.22 + 2.12i)26-s + (−0.358 + 2.62i)28-s − 1.24i·29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.790 − 0.612i)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 + (0.213 − 0.123i)17-s + (0.201 + 0.116i)19-s − 0.639i·22-s + (−0.442 + 0.766i)23-s + (−0.240 + 0.416i)26-s + (−0.0677 + 0.495i)28-s − 0.230i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.683 - 0.729i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.683 - 0.729i)$
$L(1)$  $\approx$  $0.6803508220$
$L(\frac12)$  $\approx$  $0.6803508220$
$L(\frac{3}{2})$   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 \( 1 + (2.09 + 1.62i)T \)
good11 \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 + (-0.878 + 0.507i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.878 - 0.507i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.12 - 3.67i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.24iT - 29T^{2} \)
31 \( 1 + (-4.86 + 2.80i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.13 + 4.12i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.02T + 41T^{2} \)
43 \( 1 + 8.24iT - 43T^{2} \)
47 \( 1 + (0.878 + 0.507i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.621 + 1.07i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.76 - 9.98i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.12 - 2.95i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (8.66 - 5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 10.2iT - 71T^{2} \)
73 \( 1 + (4.18 + 7.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.62 - 9.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.16iT - 83T^{2} \)
89 \( 1 + (5.19 - 9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.76T + 97T^{2} \)
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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

−8.874338256019644758105396905024, −8.147322351016893224892839188906, −7.35472027298648676042379777351, −6.92153347556923790070989043330, −5.92806897288792036647143982255, −5.41006979103571987066039643275, −4.28096335719037003699053984568, −3.55933923881666819745495990557, −2.39618119285263659760705717789, −1.03274135121225810912700855272, 0.27649738723943874449910614904, 1.65896241700584935050024241503, 2.85621654618056608942195261847, 3.26641393561378319125480958864, 4.40785975055105999973579497048, 5.35538887977348708506305178677, 6.16633415714141949806262534283, 6.86006648055251057387102711450, 7.987689578278984871137929785102, 8.464750860689275932695883654652

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