Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $-0.992 + 0.120i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)2-s + (0.0350 − 1.73i)3-s + (0.499 − 0.866i)4-s − 5-s + (−0.835 − 1.51i)6-s + (−1.17 − 2.36i)7-s − 0.999i·8-s + (−2.99 − 0.121i)9-s + (−0.866 + 0.5i)10-s − 0.751i·11-s + (−1.48 − 0.896i)12-s + (−2.72 + 1.57i)13-s + (−2.20 − 1.46i)14-s + (−0.0350 + 1.73i)15-s + (−0.5 − 0.866i)16-s + (0.433 + 0.750i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.0202 − 0.999i)3-s + (0.249 − 0.433i)4-s − 0.447·5-s + (−0.341 − 0.619i)6-s + (−0.445 − 0.895i)7-s − 0.353i·8-s + (−0.999 − 0.0405i)9-s + (−0.273 + 0.158i)10-s − 0.226i·11-s + (−0.427 − 0.258i)12-s + (−0.754 + 0.435i)13-s + (−0.589 − 0.390i)14-s + (−0.00906 + 0.447i)15-s + (−0.125 − 0.216i)16-s + (0.105 + 0.181i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.992 + 0.120i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (551, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ -0.992 + 0.120i)\)
\(L(1)\)  \(\approx\)  \(0.0829468 - 1.37549i\)
\(L(\frac12)\)  \(\approx\)  \(0.0829468 - 1.37549i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.866 + 0.5i)T \)
3 \( 1 + (-0.0350 + 1.73i)T \)
5 \( 1 + T \)
7 \( 1 + (1.17 + 2.36i)T \)
good11 \( 1 + 0.751iT - 11T^{2} \)
13 \( 1 + (2.72 - 1.57i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.433 - 0.750i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.23 - 0.713i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 5.06iT - 23T^{2} \)
29 \( 1 + (5.23 + 3.02i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.38 - 3.10i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.60 + 6.24i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.08 + 3.61i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.69 + 6.40i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.72 + 2.99i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.790 - 0.456i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.21 - 7.30i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-9.30 + 5.37i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.44 + 9.43i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.43iT - 71T^{2} \)
73 \( 1 + (-0.539 + 0.311i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.00292 + 0.00506i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.01 + 1.76i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.22 - 2.12i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.23 - 5.33i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.42763928647418035184895388674, −9.382028569881261786565349890657, −8.220731928682377653537497949388, −7.29974578758001606638381047083, −6.69993268989023967679897939235, −5.67368863422241150781556956614, −4.41929870226896205818744535194, −3.39495384802926482232704978593, −2.19886753705151887303471928536, −0.59348831546055381236240889898, 2.66061217056700177760987618225, 3.47672851252750265385211160749, 4.66283848662271530340253862428, 5.37760149347131624718492966658, 6.28488968401593176356750955139, 7.52431015587969188846468911588, 8.360943609958829146158858183927, 9.429406356523172312137676351694, 9.940325935866566029716048381443, 11.26466234990079148190514131557

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