Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.987 - 0.157i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + (−1.56 − 0.733i)3-s + 4-s + (−0.5 − 0.866i)5-s + (1.56 + 0.733i)6-s + (−2.56 − 0.658i)7-s − 8-s + (1.92 + 2.30i)9-s + (0.5 + 0.866i)10-s + (−2.23 + 3.87i)11-s + (−1.56 − 0.733i)12-s + (0.827 − 1.43i)13-s + (2.56 + 0.658i)14-s + (0.149 + 1.72i)15-s + 16-s + (1.35 + 2.34i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.905 − 0.423i)3-s + 0.5·4-s + (−0.223 − 0.387i)5-s + (0.640 + 0.299i)6-s + (−0.968 − 0.249i)7-s − 0.353·8-s + (0.641 + 0.767i)9-s + (0.158 + 0.273i)10-s + (−0.674 + 1.16i)11-s + (−0.452 − 0.211i)12-s + (0.229 − 0.397i)13-s + (0.684 + 0.176i)14-s + (0.0385 + 0.445i)15-s + 0.250·16-s + (0.328 + 0.568i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.157i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.987 - 0.157i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (121, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 630,\ (\ :1/2),\ 0.987 - 0.157i)$
$L(1)$  $\approx$  $0.555827 + 0.0441687i$
$L(\frac12)$  $\approx$  $0.555827 + 0.0441687i$
$L(\frac{3}{2})$   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 \( 1 + (1.56 + 0.733i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.56 + 0.658i)T \)
good11 \( 1 + (2.23 - 3.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.827 + 1.43i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.35 - 2.34i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.827 + 1.43i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.474 + 0.821i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.07 - 3.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.00T + 31T^{2} \)
37 \( 1 + (-4.87 + 8.44i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.970 + 1.68i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.50 - 4.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 1.01T + 47T^{2} \)
53 \( 1 + (-6.65 - 11.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.84T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 - 0.773T + 67T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + (-0.166 - 0.287i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 9.92T + 79T^{2} \)
83 \( 1 + (-1.60 - 2.78i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.86 - 4.96i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.85 + 4.95i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.41931807267123369531242423065, −10.00338752300062786997470425315, −8.938038874286495827538552209373, −7.76293361327596827221061451836, −7.20657101386447479929037526865, −6.25250677981504878664083655186, −5.33375024781139911480343178199, −4.11942849876352838137586156131, −2.47802009485665791030683572322, −0.886672884201509024030991763972, 0.61286076689001963247578829528, 2.80028010869062563103418064514, 3.81785246438264417960250104038, 5.34772430522949831137593777097, 6.17754490325300719171157517221, 6.86331825747348119968889301748, 8.004657412542338747093800983237, 8.981025080599647663483617760996, 9.986599611903247024128002267176, 10.32205876077503928651293498990

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