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Properties

Label 2-462-77.6-c1-0-6
Degree $2$
Conductor $462$
Sign $0.839 - 0.543i$
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  + (−0.951 + 0.309i)2-s + (−0.587 + 0.809i)3-s + (0.809 − 0.587i)4-s + (3.76 + 1.22i)5-s + (0.309 − 0.951i)6-s + (2.55 − 0.690i)7-s + (−0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s − 3.95·10-s + (3.31 + 0.0867i)11-s + 0.999i·12-s + (−0.469 − 1.44i)13-s + (−2.21 + 1.44i)14-s + (−3.20 + 2.32i)15-s + (0.309 − 0.951i)16-s + (1.91 − 5.90i)17-s + ⋯
L(s)  = 1  + (−0.672 + 0.218i)2-s + (−0.339 + 0.467i)3-s + (0.404 − 0.293i)4-s + (1.68 + 0.547i)5-s + (0.126 − 0.388i)6-s + (0.965 − 0.261i)7-s + (−0.207 + 0.286i)8-s + (−0.103 − 0.317i)9-s − 1.25·10-s + (0.999 + 0.0261i)11-s + 0.288i·12-s + (−0.130 − 0.400i)13-s + (−0.592 + 0.386i)14-s + (−0.826 + 0.600i)15-s + (0.0772 − 0.237i)16-s + (0.465 − 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (391, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.32432 + 0.390968i\)
\(L(\frac12)\) \(\approx\) \(1.32432 + 0.390968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.951 - 0.309i)T \)
3 \( 1 + (0.587 - 0.809i)T \)
7 \( 1 + (-2.55 + 0.690i)T \)
11 \( 1 + (-3.31 - 0.0867i)T \)
good5 \( 1 + (-3.76 - 1.22i)T + (4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.469 + 1.44i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-1.91 + 5.90i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.30 + 3.85i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 1.15T + 23T^{2} \)
29 \( 1 + (0.0126 + 0.0174i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (5.54 - 1.80i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (5.14 - 3.73i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-6.06 - 4.40i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 8.11iT - 43T^{2} \)
47 \( 1 + (0.361 - 0.497i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.09 - 6.43i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.63 + 5.00i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.52 + 10.8i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 + (2.09 - 6.43i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (12.2 - 8.93i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (11.7 - 3.83i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (1.65 - 5.08i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 0.0879iT - 89T^{2} \)
97 \( 1 + (0.719 - 0.233i)T + (78.4 - 57.0i)T^{2} \)
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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

−10.93808987096936193379151000393, −10.13898768236901287830942119752, −9.441990576678458028605173457847, −8.735235624520398865509896947168, −7.32467992586969351973539356155, −6.50072892297969590994285287251, −5.60258683031965811307867470696, −4.64961748146678501028137443864, −2.75708793270275035999620209578, −1.44276452430658972434058488143, 1.60988134916208083200040365786, 1.94454003826494480442524862424, 4.18294217591428344126137781613, 5.69329466988440702853098056560, 6.10112062531735545554125718712, 7.34658434729021624103905731297, 8.655540484019953723494536527170, 8.944019741716817012929577648696, 10.20457008710085099214192810863, 10.70228127842844272422778111140

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