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Properties

Label 2-462-33.29-c1-0-10
Degree $2$
Conductor $462$
Sign $0.927 - 0.373i$
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  + (−0.809 − 0.587i)2-s + (1.72 − 0.126i)3-s + (0.309 + 0.951i)4-s + (2.46 + 3.39i)5-s + (−1.47 − 0.913i)6-s + (−0.951 + 0.309i)7-s + (0.309 − 0.951i)8-s + (2.96 − 0.436i)9-s − 4.19i·10-s + (−0.214 − 3.30i)11-s + (0.653 + 1.60i)12-s + (−2.42 + 3.33i)13-s + (0.951 + 0.309i)14-s + (4.68 + 5.54i)15-s + (−0.809 + 0.587i)16-s + (3.28 − 2.38i)17-s + ⋯
L(s)  = 1  + (−0.572 − 0.415i)2-s + (0.997 − 0.0729i)3-s + (0.154 + 0.475i)4-s + (1.10 + 1.51i)5-s + (−0.600 − 0.372i)6-s + (−0.359 + 0.116i)7-s + (0.109 − 0.336i)8-s + (0.989 − 0.145i)9-s − 1.32i·10-s + (−0.0646 − 0.997i)11-s + (0.188 + 0.462i)12-s + (−0.672 + 0.926i)13-s + (0.254 + 0.0825i)14-s + (1.20 + 1.43i)15-s + (−0.202 + 0.146i)16-s + (0.796 − 0.578i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.373i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.927 - 0.373i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.927 - 0.373i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66973 + 0.323121i\)
\(L(\frac12)\) \(\approx\) \(1.66973 + 0.323121i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.809 + 0.587i)T \)
3 \( 1 + (-1.72 + 0.126i)T \)
7 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (0.214 + 3.30i)T \)
good5 \( 1 + (-2.46 - 3.39i)T + (-1.54 + 4.75i)T^{2} \)
13 \( 1 + (2.42 - 3.33i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-3.28 + 2.38i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.43 + 0.465i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.69iT - 23T^{2} \)
29 \( 1 + (-1.71 - 5.27i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.24 + 2.35i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.472 + 1.45i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.67 + 5.14i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 10.5iT - 43T^{2} \)
47 \( 1 + (2.85 + 0.928i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-0.246 + 0.339i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-14.2 + 4.64i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.17 - 1.61i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + (-0.0399 - 0.0550i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-1.18 + 0.384i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.09 - 5.63i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.25 - 0.914i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 14.2iT - 89T^{2} \)
97 \( 1 + (-2.16 - 1.57i)T + (29.9 + 92.2i)T^{2} \)
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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

−10.80930880980126504150594692563, −10.06098127799032231148351049616, −9.447988090488411782753210006419, −8.676570514458246229233938424402, −7.30138445600488136828366506193, −6.88136354180377938830894729943, −5.63629345842486544806843047453, −3.62875864609675641067597081649, −2.83502068199261724775577835194, −1.93501893792548022009681651658, 1.31700848529138253782681208641, 2.51866457318386890809013171249, 4.37751579433038306271260223181, 5.28188552841569482230951402939, 6.39735463372755846033784893804, 7.70809996591123178595315886077, 8.324315263123040052943406840607, 9.239354859507423539165302370627, 9.914715690646655598846759598457, 10.25284867620400408471736442369

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