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Properties

Label 2-462-231.32-c1-0-22
Degree $2$
Conductor $462$
Sign $0.890 + 0.455i$
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  + (0.5 + 0.866i)2-s + (−1.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (2.63 − 1.52i)5-s − 1.73i·6-s + (−2.63 − 0.209i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (2.63 + 1.52i)10-s + (3.13 + 1.07i)11-s + (1.49 − 0.866i)12-s − 6.09i·13-s + (−1.13 − 2.38i)14-s − 5.27·15-s + (−0.5 − 0.866i)16-s + (2.13 − 3.70i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.866 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (1.17 − 0.680i)5-s − 0.707i·6-s + (−0.996 − 0.0791i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (0.834 + 0.481i)10-s + (0.945 + 0.324i)11-s + (0.433 − 0.250i)12-s − 1.68i·13-s + (−0.303 − 0.638i)14-s − 1.36·15-s + (−0.125 − 0.216i)16-s + (0.518 − 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.890 + 0.455i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.890 + 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35626 - 0.326705i\)
\(L(\frac12)\) \(\approx\) \(1.35626 - 0.326705i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (1.5 + 0.866i)T \)
7 \( 1 + (2.63 + 0.209i)T \)
11 \( 1 + (-3.13 - 1.07i)T \)
good5 \( 1 + (-2.63 + 1.52i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 6.09iT - 13T^{2} \)
17 \( 1 + (-2.13 + 3.70i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.86 - 1.07i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.41 + 3.70i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-1.36 + 2.35i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.137 + 0.238i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.54T + 41T^{2} \)
43 \( 1 - 5.61iT - 43T^{2} \)
47 \( 1 + (9.41 - 5.43i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.18 + 4.14i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.5 - 0.866i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6 - 3.46i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.27 - 5.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.61iT - 71T^{2} \)
73 \( 1 + (-10.5 - 6.09i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.63 - 3.25i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.72T + 83T^{2} \)
89 \( 1 + (5.27 - 3.04i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.54T + 97T^{2} \)
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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

−10.99755277620048996322941332612, −9.917859089256468700536682190518, −9.341136302786022501201520880594, −8.071782047232941215529693891259, −6.95987871615479347008302648112, −6.18634079693378358559496130663, −5.53703346840373901460473597738, −4.62012884148821899606724632983, −2.87288459780854187868838357520, −0.957133372024537941891006190336, 1.61402658895754801508188353261, 3.20215504992628407221056189612, 4.20777083617075768569707484115, 5.51181859897001382206390839456, 6.47570865736353871190525546865, 6.68672756776638948622203020913, 9.181163741202590620530679354169, 9.406669633805611887273454843633, 10.35841579321544085459856834295, 11.04467216827754425996624464124

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