↑

Properties

Label 2-62-31.30-c2-0-3
Degree $2$
Conductor $62$
Sign $-0.787 + 0.616i$
Analytic cond. $1.68937$
Root an. cond. $1.29976$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s − 4.10i·3-s + 2.00·4-s − 5.82·5-s + 5.80i·6-s − 4.41·7-s − 2.82·8-s − 7.82·9-s + 8.24·10-s − 19.8i·11-s − 8.20i·12-s + 6.50i·13-s + 6.24·14-s + 23.9i·15-s + 4.00·16-s − 13.3i·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.36i·3-s + 0.500·4-s − 1.16·5-s + 0.966i·6-s − 0.630·7-s − 0.353·8-s − 0.869·9-s + 0.824·10-s − 1.80i·11-s − 0.683i·12-s + 0.500i·13-s + 0.445·14-s + 1.59i·15-s + 0.250·16-s − 0.782i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(62\)    =    \(2 \cdot 31\)
Sign: $-0.787 + 0.616i$
Analytic conductor: \(1.68937\)
Root analytic conductor: \(1.29976\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{62} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 62,\ (\ :1),\ -0.787 + 0.616i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.187005 - 0.542450i\)
\(L(\frac12)\) \(\approx\) \(0.187005 - 0.542450i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
31 \( 1 + (-24.4 + 19.1i)T \)
good3 \( 1 + 4.10iT - 9T^{2} \)
5 \( 1 + 5.82T + 25T^{2} \)
7 \( 1 + 4.41T + 49T^{2} \)
11 \( 1 + 19.8iT - 121T^{2} \)
13 \( 1 - 6.50iT - 169T^{2} \)
17 \( 1 + 13.3iT - 289T^{2} \)
19 \( 1 - 30.2T + 361T^{2} \)
23 \( 1 - 19.1iT - 529T^{2} \)
29 \( 1 + 6.50iT - 841T^{2} \)
37 \( 1 + 59.4iT - 1.36e3T^{2} \)
41 \( 1 - 16.6T + 1.68e3T^{2} \)
43 \( 1 - 65.5iT - 1.84e3T^{2} \)
47 \( 1 + 67.7T + 2.20e3T^{2} \)
53 \( 1 - 48.4iT - 2.80e3T^{2} \)
59 \( 1 - 56.2T + 3.48e3T^{2} \)
61 \( 1 - 4.51iT - 3.72e3T^{2} \)
67 \( 1 - 21.0T + 4.48e3T^{2} \)
71 \( 1 - 11.2T + 5.04e3T^{2} \)
73 \( 1 + 25.1iT - 5.32e3T^{2} \)
79 \( 1 + 36.0iT - 6.24e3T^{2} \)
83 \( 1 + 159. iT - 6.88e3T^{2} \)
89 \( 1 - 143. iT - 7.92e3T^{2} \)
97 \( 1 - 27.7T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.09558527847465981708419388576, −13.15191099298942909054680472866, −11.75190787154576330088508748666, −11.39855552055676216446435319721, −9.409283162587213060680042154039, −8.063138948406369496430010864248, −7.34917933165867049137681106723, −6.08617506241363159642278533489, −3.21256513685453433484548821373, −0.68275419580956173288454684441, 3.46001567064025290659703948421, 4.82402789990281038419836873184, 7.01070514718768674965743759441, 8.295956691131380786643545650414, 9.744090012185641192625451458389, 10.22696530130259016811400663173, 11.60453533795933769517269544616, 12.61782999078207141927354309080, 14.74424739278741234944950296522, 15.53455766560669817251212949366

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