Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.304 + 0.952i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.304 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.304 + 0.952i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.304 + 0.952i)$
$L(1)$  $\approx$  $0.7055981761$
$L(\frac12)$  $\approx$  $0.7055981761$
$L(\frac{3}{2})$   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

−8.472444328731528936972253050397, −7.67176155168424917463927296473, −6.93636221145432955283146100685, −6.16926822578775397791321243231, −5.76288149999720095000244051071, −4.64764356647596322082045202879, −3.84465630432460655073277839728, −3.12272225191628554478085169023, −1.95144812555517185202061029711, −0.19879183478036233114307860070, 1.20634510965884676734576158392, 2.40511549422582034219912130728, 3.30316851030114369701280617593, 3.97407378882131715874970118117, 4.86718849397665825430853608251, 5.76289548777557837256028587058, 6.70638606941164625639632269620, 6.91563109340767833330267447825, 8.494429531294308295002245766776, 8.778207797278474468265299079067

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