Properties

Label 2-50-5.4-c7-0-9
Degree $2$
Conductor $50$
Sign $0.447 - 0.894i$
Analytic cond. $15.6192$
Root an. cond. $3.95211$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8i·2-s − 87i·3-s − 64·4-s − 696·6-s + 1.36e3i·7-s + 512i·8-s − 5.38e3·9-s − 1.08e3·11-s + 5.56e3i·12-s + 5.46e3i·13-s + 1.09e4·14-s + 4.09e3·16-s − 2.52e4i·17-s + 4.30e4i·18-s − 3.34e4·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.86i·3-s − 0.5·4-s − 1.31·6-s + 1.50i·7-s + 0.353i·8-s − 2.46·9-s − 0.245·11-s + 0.930i·12-s + 0.690i·13-s + 1.06·14-s + 0.250·16-s − 1.24i·17-s + 1.74i·18-s − 1.11·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
Analytic conductor: \(15.6192\)
Root analytic conductor: \(3.95211\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{50} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 50,\ (\ :7/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 8iT \)
5 \( 1 \)
good3 \( 1 + 87iT - 2.18e3T^{2} \)
7 \( 1 - 1.36e3iT - 8.23e5T^{2} \)
11 \( 1 + 1.08e3T + 1.94e7T^{2} \)
13 \( 1 - 5.46e3iT - 6.27e7T^{2} \)
17 \( 1 + 2.52e4iT - 4.10e8T^{2} \)
19 \( 1 + 3.34e4T + 8.93e8T^{2} \)
23 \( 1 - 5.83e3iT - 3.40e9T^{2} \)
29 \( 1 + 1.25e5T + 1.72e10T^{2} \)
31 \( 1 + 7.37e4T + 2.75e10T^{2} \)
37 \( 1 - 3.95e5iT - 9.49e10T^{2} \)
41 \( 1 + 2.26e4T + 1.94e11T^{2} \)
43 \( 1 - 1.00e5iT - 2.71e11T^{2} \)
47 \( 1 + 1.14e6iT - 5.06e11T^{2} \)
53 \( 1 + 3.54e5iT - 1.17e12T^{2} \)
59 \( 1 + 1.09e6T + 2.48e12T^{2} \)
61 \( 1 + 4.22e5T + 3.14e12T^{2} \)
67 \( 1 + 2.55e6iT - 6.06e12T^{2} \)
71 \( 1 + 2.28e6T + 9.09e12T^{2} \)
73 \( 1 - 6.37e6iT - 1.10e13T^{2} \)
79 \( 1 - 2.01e6T + 1.92e13T^{2} \)
83 \( 1 - 7.97e6iT - 2.71e13T^{2} \)
89 \( 1 + 2.18e6T + 4.42e13T^{2} \)
97 \( 1 - 5.82e6iT - 8.07e13T^{2} \)
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

−12.88123770649312800336407759976, −12.03922250895695304923172781398, −11.35896328845855456431427803407, −9.195241619677209107294753836989, −8.228167752248924171994145447757, −6.75684436994422189009079076497, −5.44803315489306644200442776995, −2.68779083382970985842253190328, −1.77821187710375870569819037595, 0, 3.65652613527578717941828432631, 4.49325139267634280104137626393, 5.92674755316910020425576363470, 7.80295365569050377995688990454, 9.102756940554523753054050199878, 10.42354110184372121938524836929, 10.77384414669595965826543634727, 13.02720046326127208991362982573, 14.37608967902891957404635912309

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