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Properties

Label 2-462-231.32-c1-0-16
Degree $2$
Conductor $462$
Sign $-0.207 - 0.978i$
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 + (1.13 − 0.656i)5-s + 1.73i·6-s + (−1.13 + 2.38i)7-s − 0.999·8-s + (1.5 + 2.59i)9-s + (1.13 + 0.656i)10-s + (−2.5 + 2.17i)11-s + (−1.49 + 0.866i)12-s − 2.62i·13-s + (−2.63 + 0.209i)14-s + 2.27·15-s + (−0.5 − 0.866i)16-s + (−1.63 + 2.83i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.866 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.508 − 0.293i)5-s + 0.707i·6-s + (−0.429 + 0.902i)7-s − 0.353·8-s + (0.5 + 0.866i)9-s + (0.359 + 0.207i)10-s + (−0.753 + 0.657i)11-s + (−0.433 + 0.250i)12-s − 0.728i·13-s + (−0.704 + 0.0559i)14-s + 0.587·15-s + (−0.125 − 0.216i)16-s + (−0.397 + 0.687i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.207 - 0.978i$
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: \((2,\ 462,\ (\ :1/2),\ -0.207 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.36621 + 1.68676i\)
\(L(\frac12)\) \(\approx\) \(1.36621 + 1.68676i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.13 - 2.38i)T \)
11 \( 1 + (2.5 - 2.17i)T \)
good5 \( 1 + (-1.13 + 0.656i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 2.62iT - 13T^{2} \)
17 \( 1 + (1.63 - 2.83i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.63 + 3.25i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.91 + 2.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + (-5.13 + 8.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.63 - 6.30i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 6.54T + 41T^{2} \)
43 \( 1 + 9.97iT - 43T^{2} \)
47 \( 1 + (1.91 - 1.10i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.6 + 6.74i)T + (26.5 + 45.8i)T^{2} \)
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 \( 1 + (-4.27 + 7.40i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.97iT - 71T^{2} \)
73 \( 1 + (-4.54 - 2.62i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.86 + 1.07i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (2.27 - 1.31i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−11.29023923521494738761536615101, −9.980225398908358505497073025628, −9.500592994200721822175942004956, −8.536006969556391609201558579826, −7.81942242233366343272188811582, −6.65874049785282579392640983858, −5.40777516291165696685786348317, −4.81975741474389743653839110795, −3.30920562974616099709898504477, −2.35357867605031612699881086198, 1.23777726211186728535550984919, 2.75323262238177310536959935458, 3.47810205278236330898581687146, 4.83136391387339151168798874534, 6.22463520039020577405745043501, 7.09340396183370990095555520590, 8.056650964202118662149530323550, 9.267288487730796905049777032266, 9.855512276256295074045633380769, 10.75927033401738723029558366237

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