Properties

Label 2-50-5.4-c7-0-2
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 $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 8i·2-s + 43i·3-s − 64·4-s + 344·6-s − 974i·7-s + 512i·8-s + 338·9-s + 87·11-s − 2.75e3i·12-s + 1.48e4i·13-s − 7.79e3·14-s + 4.09e3·16-s + 3.55e4i·17-s − 2.70e3i·18-s − 2.06e4·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.919i·3-s − 0.5·4-s + 0.650·6-s − 1.07i·7-s + 0.353i·8-s + 0.154·9-s + 0.0197·11-s − 0.459i·12-s + 1.87i·13-s − 0.758·14-s + 0.250·16-s + 1.75i·17-s − 0.109i·18-s − 0.689·19-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: \(0\)
Selberg data: \((2,\ 50,\ (\ :7/2),\ 0.447 - 0.894i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.23018 + 0.760293i\)
\(L(\frac12)\) \(\approx\) \(1.23018 + 0.760293i\)
\(L(\frac{9}{2})\) 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 \( 1 - 43iT - 2.18e3T^{2} \)
7 \( 1 + 974iT - 8.23e5T^{2} \)
11 \( 1 - 87T + 1.94e7T^{2} \)
13 \( 1 - 1.48e4iT - 6.27e7T^{2} \)
17 \( 1 - 3.55e4iT - 4.10e8T^{2} \)
19 \( 1 + 2.06e4T + 8.93e8T^{2} \)
23 \( 1 - 2.22e4iT - 3.40e9T^{2} \)
29 \( 1 - 5.76e3T + 1.72e10T^{2} \)
31 \( 1 - 3.02e5T + 2.75e10T^{2} \)
37 \( 1 - 1.99e5iT - 9.49e10T^{2} \)
41 \( 1 + 6.68e5T + 1.94e11T^{2} \)
43 \( 1 + 1.43e5iT - 2.71e11T^{2} \)
47 \( 1 - 3.38e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.09e6iT - 1.17e12T^{2} \)
59 \( 1 - 2.13e6T + 2.48e12T^{2} \)
61 \( 1 + 1.93e6T + 3.14e12T^{2} \)
67 \( 1 - 3.34e6iT - 6.06e12T^{2} \)
71 \( 1 - 3.00e6T + 9.09e12T^{2} \)
73 \( 1 + 3.04e6iT - 1.10e13T^{2} \)
79 \( 1 + 5.48e6T + 1.92e13T^{2} \)
83 \( 1 - 5.20e6iT - 2.71e13T^{2} \)
89 \( 1 - 8.32e5T + 4.42e13T^{2} \)
97 \( 1 + 5.31e6iT - 8.07e13T^{2} \)
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

−14.16688197414197468666448117401, −13.13517790333676467404908832497, −11.69751067384261940494062828755, −10.55146985293155677211154693095, −9.864047222014032111502379294933, −8.522776225580797572459266469227, −6.68177052816759172590118862648, −4.50850863623915554992925074878, −3.82779986782812114223491764487, −1.57521835173078589149517531810, 0.60748270348491748283405106520, 2.67154715893428731868120466282, 5.08435329675783326960560016293, 6.33330573570826556039727753838, 7.59111976954572922954097766812, 8.611993939043239472038383539869, 10.09476403403871450274082065936, 11.93454144118981646687682092612, 12.81044879773764399094288245414, 13.77788002581556893532672254282

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