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Properties

Label 2-483-21.20-c1-0-42
Degree $2$
Conductor $483$
Sign $0.380 - 0.924i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
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  + 1.77i·2-s + (1.65 − 0.524i)3-s − 1.14·4-s + 3.73·5-s + (0.929 + 2.92i)6-s + (−2.02 − 1.69i)7-s + 1.52i·8-s + (2.45 − 1.73i)9-s + 6.61i·10-s − 1.46i·11-s + (−1.88 + 0.598i)12-s + 0.721i·13-s + (3.01 − 3.59i)14-s + (6.15 − 1.95i)15-s − 4.97·16-s − 4.35·17-s + ⋯
L(s)  = 1  + 1.25i·2-s + (0.953 − 0.302i)3-s − 0.570·4-s + 1.66·5-s + (0.379 + 1.19i)6-s + (−0.766 − 0.642i)7-s + 0.537i·8-s + (0.816 − 0.577i)9-s + 2.09i·10-s − 0.441i·11-s + (−0.544 + 0.172i)12-s + 0.200i·13-s + (0.805 − 0.960i)14-s + (1.58 − 0.505i)15-s − 1.24·16-s − 1.05·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.380 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.380 - 0.924i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.380 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96992 + 1.31977i\)
\(L(\frac12)\) \(\approx\) \(1.96992 + 1.31977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.65 + 0.524i)T \)
7 \( 1 + (2.02 + 1.69i)T \)
23 \( 1 + iT \)
good2 \( 1 - 1.77iT - 2T^{2} \)
5 \( 1 - 3.73T + 5T^{2} \)
11 \( 1 + 1.46iT - 11T^{2} \)
13 \( 1 - 0.721iT - 13T^{2} \)
17 \( 1 + 4.35T + 17T^{2} \)
19 \( 1 - 0.368iT - 19T^{2} \)
29 \( 1 + 2.64iT - 29T^{2} \)
31 \( 1 - 9.03iT - 31T^{2} \)
37 \( 1 + 4.69T + 37T^{2} \)
41 \( 1 + 1.09T + 41T^{2} \)
43 \( 1 + 9.97T + 43T^{2} \)
47 \( 1 - 4.98T + 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + 7.05T + 59T^{2} \)
61 \( 1 + 7.71iT - 61T^{2} \)
67 \( 1 + 3.88T + 67T^{2} \)
71 \( 1 + 12.2iT - 71T^{2} \)
73 \( 1 - 14.1iT - 73T^{2} \)
79 \( 1 + 9.18T + 79T^{2} \)
83 \( 1 - 1.74T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 - 10.2iT - 97T^{2} \)
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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

−10.79782067075870411304511617185, −9.952733648524400512907043156593, −9.073852785393944099825276928136, −8.518313510923825302629694929575, −7.22141046238510850966884844872, −6.61066984481744393018260068757, −5.97999406031355008780997436386, −4.69336107245262008018315104115, −3.06012027412308519638803073904, −1.84703541378439454555847224006, 1.87177798376988588644121076395, 2.43967665486947624807599139158, 3.45134405591780243490753113584, 4.82513843971370836657394893835, 6.12575060186950118609221096579, 7.06773860385840214237584780763, 8.770137780272749666942333387587, 9.304379679784188185038898071040, 9.993828529181325209182146199961, 10.43871082238689325768717917785

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