Properties

Label 2-462-21.5-c1-0-14
Degree $2$
Conductor $462$
Sign $-0.118 - 0.992i$
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  + (0.866 + 0.5i)2-s + (1.53 + 0.801i)3-s + (0.499 + 0.866i)4-s + (−1.79 + 3.10i)5-s + (0.928 + 1.46i)6-s + (0.833 − 2.51i)7-s + 0.999i·8-s + (1.71 + 2.46i)9-s + (−3.10 + 1.79i)10-s + (−0.866 + 0.5i)11-s + (0.0733 + 1.73i)12-s − 0.365i·13-s + (1.97 − 1.75i)14-s + (−5.24 + 3.33i)15-s + (−0.5 + 0.866i)16-s + (−1.04 − 1.81i)17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.886 + 0.462i)3-s + (0.249 + 0.433i)4-s + (−0.802 + 1.39i)5-s + (0.379 + 0.596i)6-s + (0.315 − 0.949i)7-s + 0.353i·8-s + (0.571 + 0.820i)9-s + (−0.983 + 0.567i)10-s + (−0.261 + 0.150i)11-s + (0.0211 + 0.499i)12-s − 0.101i·13-s + (0.528 − 0.469i)14-s + (−1.35 + 0.860i)15-s + (−0.125 + 0.216i)16-s + (−0.254 − 0.440i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.118 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.118 - 0.992i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.118 - 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55246 + 1.74832i\)
\(L(\frac12)\) \(\approx\) \(1.55246 + 1.74832i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 + (-1.53 - 0.801i)T \)
7 \( 1 + (-0.833 + 2.51i)T \)
11 \( 1 + (0.866 - 0.5i)T \)
good5 \( 1 + (1.79 - 3.10i)T + (-2.5 - 4.33i)T^{2} \)
13 \( 1 + 0.365iT - 13T^{2} \)
17 \( 1 + (1.04 + 1.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.84 - 3.37i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.78 + 1.03i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 3.75iT - 29T^{2} \)
31 \( 1 + (8.04 - 4.64i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.32 + 9.22i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.58T + 41T^{2} \)
43 \( 1 - 4.30T + 43T^{2} \)
47 \( 1 + (-6.39 + 11.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.13 + 1.80i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-3.85 - 6.67i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.48 - 4.32i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.98 + 5.16i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 4.43iT - 71T^{2} \)
73 \( 1 + (-4.69 + 2.70i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.30 - 5.72i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.62T + 83T^{2} \)
89 \( 1 + (7.03 - 12.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.651iT - 97T^{2} \)
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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

−11.10081507846444940370789945634, −10.56888136209990793169967613890, −9.617743146949534703393094277739, −8.231993799153881476482199760317, −7.31680899812073818629299045988, −7.20804140979948651072645290084, −5.51500473862787835037664531405, −4.11511235732799927647598567510, −3.61096643033176747377337114324, −2.48579803947393596345511728340, 1.24679011934949238860738493392, 2.64587516636778916173118427263, 3.89052772145794105026424952805, 4.88115123414947469160373438475, 5.85304945162221137212867525915, 7.36812407346970974447382692355, 8.157627784869978895090481405550, 8.985790096992054032904540874578, 9.577247787896753970523326245684, 11.26756788657083205020336521010

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