Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $-0.287 + 0.957i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−0.433 − 1.67i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.433 + 1.67i)6-s + (1.23 + 2.33i)7-s − 8-s + (−2.62 + 1.45i)9-s + (0.5 − 0.866i)10-s + (−2.00 − 3.47i)11-s + (−0.433 − 1.67i)12-s + (−0.0260 − 0.0451i)13-s + (−1.23 − 2.33i)14-s + (1.66 + 0.462i)15-s + 16-s + (3.62 − 6.27i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.250 − 0.968i)3-s + 0.5·4-s + (−0.223 + 0.387i)5-s + (0.177 + 0.684i)6-s + (0.468 + 0.883i)7-s − 0.353·8-s + (−0.874 + 0.485i)9-s + (0.158 − 0.273i)10-s + (−0.605 − 1.04i)11-s + (−0.125 − 0.484i)12-s + (−0.00723 − 0.0125i)13-s + (−0.331 − 0.624i)14-s + (0.430 + 0.119i)15-s + 0.250·16-s + (0.879 − 1.52i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.287 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.287 + 0.957i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 630,\ (\ :1/2),\ -0.287 + 0.957i)$
$L(1)$  $\approx$  $0.449915 - 0.604673i$
$L(\frac12)$  $\approx$  $0.449915 - 0.604673i$
$L(\frac{3}{2})$   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 \( 1 + T \)
3 \( 1 + (0.433 + 1.67i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-1.23 - 2.33i)T \)
good11 \( 1 + (2.00 + 3.47i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.0260 + 0.0451i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.62 + 6.27i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0260 + 0.0451i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.65 + 4.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.446 - 0.773i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.776T + 31T^{2} \)
37 \( 1 + (3.30 + 5.72i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (6.15 + 10.6i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.24 + 10.8i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.49T + 47T^{2} \)
53 \( 1 + (3.85 - 6.67i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 11.7T + 59T^{2} \)
61 \( 1 + 6.00T + 61T^{2} \)
67 \( 1 - 7.03T + 67T^{2} \)
71 \( 1 - 2.59T + 71T^{2} \)
73 \( 1 + (0.00136 - 0.00236i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 8.10T + 79T^{2} \)
83 \( 1 + (0.719 - 1.24i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.125 + 0.218i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.0494 + 0.0856i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.68811507754073662655331150785, −9.165091847835781862974560893227, −8.567783235843621598334591889574, −7.65600939379928199665105087785, −7.05116790083416221287864378964, −5.86034244271960766084868368036, −5.24011971007987124262694715975, −3.10720554267598179695983736781, −2.25010280625667741002585228036, −0.56323877189305904094591506741, 1.43755622924931846795747316274, 3.30013868776587544154278398001, 4.38190439891030272979316447518, 5.22348725958884526633224568320, 6.41455678304318441584878474169, 7.68912198001363368362827370918, 8.155884844776377568989543266620, 9.343043919446699563383518161603, 10.02208118076076866741662944130, 10.63657251065399885485004502093

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