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  :  $\chi_{3150} (1349, \cdot )$
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

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

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