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Properties

Label 2-462-231.65-c1-0-10
Degree $2$
Conductor $462$
Sign $0.957 + 0.288i$
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 + (−1.64 − 0.543i)3-s + (−0.499 − 0.866i)4-s + (1.63 + 0.941i)5-s + (−1.29 + 1.15i)6-s + (1.76 + 1.96i)7-s − 0.999·8-s + (2.40 + 1.78i)9-s + (1.63 − 0.941i)10-s + (−0.803 + 3.21i)11-s + (0.351 + 1.69i)12-s − 1.92i·13-s + (2.58 − 0.546i)14-s + (−2.16 − 2.43i)15-s + (−0.5 + 0.866i)16-s + (1.83 + 3.17i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.949 − 0.313i)3-s + (−0.249 − 0.433i)4-s + (0.729 + 0.421i)5-s + (−0.527 + 0.470i)6-s + (0.668 + 0.744i)7-s − 0.353·8-s + (0.802 + 0.596i)9-s + (0.515 − 0.297i)10-s + (−0.242 + 0.970i)11-s + (0.101 + 0.489i)12-s − 0.534i·13-s + (0.691 − 0.145i)14-s + (−0.560 − 0.628i)15-s + (−0.125 + 0.216i)16-s + (0.443 + 0.768i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.957 + 0.288i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.957 + 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.45641 - 0.214621i\)
\(L(\frac12)\) \(\approx\) \(1.45641 - 0.214621i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.64 + 0.543i)T \)
7 \( 1 + (-1.76 - 1.96i)T \)
11 \( 1 + (0.803 - 3.21i)T \)
good5 \( 1 + (-1.63 - 0.941i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 1.92iT - 13T^{2} \)
17 \( 1 + (-1.83 - 3.17i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.68 - 1.54i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.71 - 2.14i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 9.69T + 29T^{2} \)
31 \( 1 + (5.11 + 8.86i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.15 + 7.20i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.35T + 41T^{2} \)
43 \( 1 - 6.75iT - 43T^{2} \)
47 \( 1 + (4.43 + 2.55i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.07 + 1.19i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (9.50 - 5.49i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.0 - 5.80i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.56 - 9.64i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.77iT - 71T^{2} \)
73 \( 1 + (-1.82 + 1.05i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (9.73 + 5.61i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 2.34T + 83T^{2} \)
89 \( 1 + (10.7 + 6.20i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 0.0299T + 97T^{2} \)
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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

−11.14594255745282663469910736365, −10.19570006324452363957367137185, −9.723575207381020550803388212450, −8.243205720542273380452315263394, −7.17981950462511511456012989541, −5.93992673244008955171316094449, −5.44664611900464581830987513114, −4.37842165388894447769202829031, −2.60100941885032179537655786982, −1.52487627880004208249803697476, 1.09336429658957405220614639859, 3.38909147259521941466651849139, 4.95122127557766050374017032628, 5.08082182652107279656738436197, 6.38466301938592548888397866118, 7.10783613493461772716157669288, 8.333488041578013845028354836005, 9.316293083718515580821143136074, 10.26820809360294711313776040470, 11.14845702954311416198200603903

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