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Properties

Label 2-462-231.65-c1-0-23
Degree $2$
Conductor $462$
Sign $-0.176 + 0.984i$
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 + (0.351 − 1.69i)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 + (−2.75 − 1.19i)9-s + (1.63 − 0.941i)10-s + (0.803 − 3.21i)11-s + (−1.64 + 0.543i)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 + (0.202 − 0.979i)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 + (−0.917 − 0.397i)9-s + (0.515 − 0.297i)10-s + (0.242 − 0.970i)11-s + (−0.474 + 0.156i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.176 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.176 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.176 + 0.984i$
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: \((2,\ 462,\ (\ :1/2),\ -0.176 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.558708 - 0.667879i\)
\(L(\frac12)\) \(\approx\) \(0.558708 - 0.667879i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.351 + 1.69i)T \)
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

−11.22035354301961562065596768055, −9.442669661381454892301148881676, −8.761790211085425922790024384302, −7.86730960158540353775559535710, −7.55062528833461116764409062905, −6.05286795078848311551931160511, −5.49929612526831098781756713209, −3.94856835816038494067131545734, −2.30021458745459790042832345126, −0.59237402417868453630259242650, 1.95236694946960157157814271184, 3.64203108318753282576682128995, 4.11572252043492641183462761161, 5.23878520784308326494885354816, 7.04723995420145280273381943009, 7.77204202613862910139393876764, 8.790153578100863029933610118264, 9.667451352203536042440749702972, 10.44223359475414787476812753168, 11.27308299258568918425993241187

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