Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $-0.968 + 0.250i$
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 + (0.5 − 0.866i)5-s + (−2 + 1.73i)7-s − 0.999·8-s + (−0.499 − 0.866i)10-s + (−2.5 − 4.33i)11-s − 5·13-s + (0.499 + 2.59i)14-s + (−0.5 + 0.866i)16-s + (−2 − 3.46i)17-s + (3.5 − 6.06i)19-s − 0.999·20-s − 5·22-s + (0.5 − 0.866i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.223 − 0.387i)5-s + (−0.755 + 0.654i)7-s − 0.353·8-s + (−0.158 − 0.273i)10-s + (−0.753 − 1.30i)11-s − 1.38·13-s + (0.133 + 0.694i)14-s + (−0.125 + 0.216i)16-s + (−0.485 − 0.840i)17-s + (0.802 − 1.39i)19-s − 0.223·20-s − 1.06·22-s + (0.104 − 0.180i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.968 + 0.250i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (541, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ -0.968 + 0.250i)\)
\(L(1)\)  \(\approx\)  \(0.121192 - 0.951022i\)
\(L(\frac12)\)  \(\approx\)  \(0.121192 - 0.951022i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2 - 1.73i)T \)
good11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5T + 13T^{2} \)
17 \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 5T + 41T^{2} \)
43 \( 1 - 12T + 43T^{2} \)
47 \( 1 + (5.5 - 9.52i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2 - 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-7 + 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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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

−10.18428091831982305473912672176, −9.348084258811557264347194738507, −8.824237940396154554875506103122, −7.53550592955590719540367728156, −6.40845125194464362529193886438, −5.36720858753066298400408584054, −4.76693209639527399363167283380, −3.08816430600219330754148544304, −2.53113072379842688939792474628, −0.43523103010652206430414709348, 2.23024409082416265018490320927, 3.53400530488337618788129251009, 4.59177878133704356505154841170, 5.58756348665567641289927285318, 6.65236343992326758403492498746, 7.36647492212686621800083458975, 7.987044870672589076941015267156, 9.543021933622657216761220812249, 9.942536779318481670520373852285, 10.78127876532032434845701648063

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