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Properties

Label 2-462-33.29-c1-0-2
Degree $2$
Conductor $462$
Sign $0.309 - 0.950i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 − 0.587i)2-s + (−0.911 − 1.47i)3-s + (0.309 + 0.951i)4-s + (2.04 + 2.81i)5-s + (−0.128 + 1.72i)6-s + (0.951 − 0.309i)7-s + (0.309 − 0.951i)8-s + (−1.33 + 2.68i)9-s − 3.47i·10-s + (−2.97 + 1.46i)11-s + (1.11 − 1.32i)12-s + (−2.76 + 3.80i)13-s + (−0.951 − 0.309i)14-s + (2.28 − 5.57i)15-s + (−0.809 + 0.587i)16-s + (−5.89 + 4.28i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (−0.526 − 0.850i)3-s + (0.154 + 0.475i)4-s + (0.913 + 1.25i)5-s + (−0.0524 + 0.705i)6-s + (0.359 − 0.116i)7-s + (0.109 − 0.336i)8-s + (−0.446 + 0.894i)9-s − 1.09i·10-s + (−0.897 + 0.441i)11-s + (0.323 − 0.381i)12-s + (−0.767 + 1.05i)13-s + (−0.254 − 0.0825i)14-s + (0.588 − 1.43i)15-s + (−0.202 + 0.146i)16-s + (−1.43 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.309 - 0.950i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.309 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.545627 + 0.395999i\)
\(L(\frac12)\) \(\approx\) \(0.545627 + 0.395999i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (0.911 + 1.47i)T \)
7 \( 1 + (-0.951 + 0.309i)T \)
11 \( 1 + (2.97 - 1.46i)T \)
good5 \( 1 + (-2.04 - 2.81i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (2.76 - 3.80i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (5.89 - 4.28i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.19 + 0.711i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + 0.430iT - 23T^{2} \)
29 \( 1 + (0.591 + 1.82i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.01 + 0.740i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.840 - 2.58i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.21 + 6.82i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 11.3iT - 43T^{2} \)
47 \( 1 + (-10.3 - 3.36i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-4.44 + 6.11i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-7.64 + 2.48i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-5.23 - 7.20i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 3.79T + 67T^{2} \)
71 \( 1 + (8.84 + 12.1i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.238 + 0.0775i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.38 + 6.03i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (12.9 - 9.42i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 14.0iT - 89T^{2} \)
97 \( 1 + (-3.63 - 2.64i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.95414572848856293165102849291, −10.62007837353086696136097778865, −9.656916965809912570281697460139, −8.489070913809769282508633656508, −7.37061815458346845361375561245, −6.78282195835880554363042071965, −5.92021275819516984468619704722, −4.47777569988703511144666526343, −2.44927392193545819503000514254, −2.02088351409436581377975184849, 0.49774029387315125246082611245, 2.45957397977845310523457676637, 4.52533011906702283325491409921, 5.32366738278673513913096104792, 5.79127458270588610652610207906, 7.23754328401236416873918817628, 8.576313008183931775142565441378, 8.974927360575584697724018757627, 9.962803857041476129825662088901, 10.56441801359971419959797178324

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