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Properties

Label 2-462-231.32-c1-0-17
Degree $2$
Conductor $462$
Sign $0.975 - 0.217i$
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 + (0.783 + 1.54i)3-s + (−0.499 + 0.866i)4-s + (3.23 − 1.86i)5-s + (0.946 − 1.45i)6-s + (0.830 + 2.51i)7-s + 0.999·8-s + (−1.77 + 2.42i)9-s + (−3.23 − 1.86i)10-s + (1.70 + 2.84i)11-s + (−1.72 − 0.0938i)12-s − 4.64i·13-s + (1.76 − 1.97i)14-s + (5.42 + 3.53i)15-s + (−0.5 − 0.866i)16-s + (0.460 − 0.798i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.452 + 0.891i)3-s + (−0.249 + 0.433i)4-s + (1.44 − 0.835i)5-s + (0.386 − 0.592i)6-s + (0.313 + 0.949i)7-s + 0.353·8-s + (−0.590 + 0.806i)9-s + (−1.02 − 0.590i)10-s + (0.515 + 0.856i)11-s + (−0.499 − 0.0270i)12-s − 1.28i·13-s + (0.470 − 0.527i)14-s + (1.39 + 0.912i)15-s + (−0.125 − 0.216i)16-s + (0.111 − 0.193i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.217i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.217i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.975 - 0.217i$
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: \((2,\ 462,\ (\ :1/2),\ 0.975 - 0.217i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68936 + 0.186298i\)
\(L(\frac12)\) \(\approx\) \(1.68936 + 0.186298i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.783 - 1.54i)T \)
7 \( 1 + (-0.830 - 2.51i)T \)
11 \( 1 + (-1.70 - 2.84i)T \)
good5 \( 1 + (-3.23 + 1.86i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 4.64iT - 13T^{2} \)
17 \( 1 + (-0.460 + 0.798i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.14 - 2.39i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.94 - 2.85i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.73T + 29T^{2} \)
31 \( 1 + (-1.26 + 2.19i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.29 + 2.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4.10T + 41T^{2} \)
43 \( 1 - 2.19iT - 43T^{2} \)
47 \( 1 + (-4.96 + 2.86i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.30 - 1.90i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.65 + 4.42i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.52 + 3.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.390 + 0.676i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.4iT - 71T^{2} \)
73 \( 1 + (9.35 + 5.40i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (14.9 - 8.60i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 12.6T + 83T^{2} \)
89 \( 1 + (7.44 - 4.30i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.1T + 97T^{2} \)
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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

−10.72388064623545797932289082746, −9.962585624431841298795220622403, −9.492391511990574635987988490612, −8.665100029605658326919674221246, −7.998260152848055122423656707678, −6.03547886644053336951439836877, −5.25932368434590347415756832639, −4.29032304126942293704790021812, −2.69908204921776522728594603406, −1.78738453639444989482502610434, 1.36236280838048529813506242973, 2.55417509591986010293799999770, 4.20147815722413025667094473826, 5.91835390310905962310764171025, 6.57699177637803757267161207955, 7.04074035133243929142104493742, 8.338934051126609561306122811890, 9.028938436776801687467560234896, 10.05597919847715410283611817832, 10.78352392061811593048613397237

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