Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.796 - 0.604i$
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 + (−2.61 − 0.397i)7-s + 0.999·8-s + (−0.429 + 0.248i)11-s + 2.74·13-s + (1.65 − 2.06i)14-s + (−0.5 + 0.866i)16-s + (3.16 − 1.82i)17-s + (−3.12 − 1.80i)19-s − 0.496i·22-s + (−3.21 + 5.56i)23-s + (−1.37 + 2.37i)26-s + (0.963 + 2.46i)28-s − 8.87i·29-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.988 − 0.150i)7-s + 0.353·8-s + (−0.129 + 0.0748i)11-s + 0.761·13-s + (0.441 − 0.552i)14-s + (−0.125 + 0.216i)16-s + (0.768 − 0.443i)17-s + (−0.716 − 0.413i)19-s − 0.105i·22-s + (−0.669 + 1.16i)23-s + (−0.269 + 0.466i)26-s + (0.182 + 0.465i)28-s − 1.64i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.796 - 0.604i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.796 - 0.604i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.796 - 0.604i)$
$L(1)$  $\approx$  $0.6479494460$
$L(\frac12)$  $\approx$  $0.6479494460$
$L(\frac{3}{2})$   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 \( 1 + (2.61 + 0.397i)T \)
good11 \( 1 + (0.429 - 0.248i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.74T + 13T^{2} \)
17 \( 1 + (-3.16 + 1.82i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.12 + 1.80i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.21 - 5.56i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.87iT - 29T^{2} \)
31 \( 1 + (6.90 - 3.98i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.98 - 1.14i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 2.22T + 41T^{2} \)
43 \( 1 - 2.22iT - 43T^{2} \)
47 \( 1 + (-5.66 - 3.27i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.88 - 6.72i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.05 + 5.28i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.24 - 1.87i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.08 - 4.08i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + (-6.53 - 11.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.44 - 7.69i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.79iT - 83T^{2} \)
89 \( 1 + (-0.743 + 1.28i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.05T + 97T^{2} \)
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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

−9.081373395235463327588507090907, −8.107490680736594390769117474892, −7.51219251409807124935967778334, −6.75187687975220582716539366730, −5.99054135644848483094994691910, −5.49436584334999311843495449342, −4.27306484259025456690861124352, −3.56382235870499748127590166850, −2.45688266444902407102400720523, −1.06949613114812199820613218275, 0.25902579465038281154963104098, 1.62563148943408353740758203468, 2.65964951326771132947509720813, 3.58788817656539528689838523492, 4.11908577311951512536680976573, 5.43544546831927095362299085367, 6.11020518636700744209865620998, 6.90766696723125998555899419850, 7.78937652391547221051937068493, 8.621393901625395743786080876279

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