Properties

Label 2-1-1.1-c75-0-3
Degree $2$
Conductor $1$
Sign $1$
Analytic cond. $35.6228$
Root an. cond. $5.96848$
Motivic weight $75$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 1.24e11·2-s + 1.35e17·3-s − 2.21e22·4-s + 1.80e26·5-s − 1.68e28·6-s + 2.76e31·7-s + 7.49e33·8-s − 5.90e35·9-s − 2.26e37·10-s + 7.60e38·11-s − 2.99e39·12-s + 2.57e41·13-s − 3.45e42·14-s + 2.44e43·15-s − 9.94e43·16-s − 1.39e46·17-s + 7.37e46·18-s − 1.16e48·19-s − 4.00e48·20-s + 3.73e48·21-s − 9.50e49·22-s + 9.81e50·23-s + 1.01e51·24-s + 6.26e51·25-s − 3.22e52·26-s − 1.61e53·27-s − 6.11e53·28-s + ⋯
L(s)  = 1  − 0.643·2-s + 0.173·3-s − 0.586·4-s + 1.11·5-s − 0.111·6-s + 0.562·7-s + 1.02·8-s − 0.969·9-s − 0.715·10-s + 0.674·11-s − 0.101·12-s + 0.434·13-s − 0.361·14-s + 0.192·15-s − 0.0697·16-s − 1.00·17-s + 0.623·18-s − 1.29·19-s − 0.652·20-s + 0.0974·21-s − 0.433·22-s + 0.845·23-s + 0.176·24-s + 0.236·25-s − 0.279·26-s − 0.341·27-s − 0.329·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & \,\Lambda(76-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+75/2) \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(35.6228\)
Root analytic conductor: \(5.96848\)
Motivic weight: \(75\)
Rational: no
Arithmetic: yes
Character: $\chi_{1} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1,\ (\ :75/2),\ 1)\)

Particular Values

\(L(38)\) \(\approx\) \(1.610368071\)
\(L(\frac12)\) \(\approx\) \(1.610368071\)
\(L(\frac{77}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 1.24e11T + 3.77e22T^{2} \)
3 \( 1 - 1.35e17T + 6.08e35T^{2} \)
5 \( 1 - 1.80e26T + 2.64e52T^{2} \)
7 \( 1 - 2.76e31T + 2.41e63T^{2} \)
11 \( 1 - 7.60e38T + 1.27e78T^{2} \)
13 \( 1 - 2.57e41T + 3.51e83T^{2} \)
17 \( 1 + 1.39e46T + 1.92e92T^{2} \)
19 \( 1 + 1.16e48T + 8.06e95T^{2} \)
23 \( 1 - 9.81e50T + 1.34e102T^{2} \)
29 \( 1 - 3.47e54T + 4.78e109T^{2} \)
31 \( 1 - 1.19e56T + 7.11e111T^{2} \)
37 \( 1 - 1.03e59T + 4.12e117T^{2} \)
41 \( 1 - 4.07e60T + 9.09e120T^{2} \)
43 \( 1 - 2.20e61T + 3.23e122T^{2} \)
47 \( 1 + 4.37e62T + 2.55e125T^{2} \)
53 \( 1 - 7.68e64T + 2.09e129T^{2} \)
59 \( 1 + 3.79e66T + 6.51e132T^{2} \)
61 \( 1 - 5.02e66T + 7.93e133T^{2} \)
67 \( 1 - 3.39e68T + 9.02e136T^{2} \)
71 \( 1 + 3.17e69T + 6.98e138T^{2} \)
73 \( 1 - 7.11e69T + 5.61e139T^{2} \)
79 \( 1 - 9.47e70T + 2.09e142T^{2} \)
83 \( 1 + 1.43e71T + 8.52e143T^{2} \)
89 \( 1 - 9.94e72T + 1.60e146T^{2} \)
97 \( 1 + 7.42e73T + 1.01e149T^{2} \)
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

−17.08849976426933116417692103830, −14.47524515780867517424241644882, −13.33854408255827387586920177614, −10.95628709701300446889046117978, −9.340907332054442788083289369497, −8.375895378057554883273929259773, −6.16233581584241431351625569540, −4.48881368106090201687271821890, −2.27943924184356855781281266347, −0.874762885394372842178720411755, 0.874762885394372842178720411755, 2.27943924184356855781281266347, 4.48881368106090201687271821890, 6.16233581584241431351625569540, 8.375895378057554883273929259773, 9.340907332054442788083289369497, 10.95628709701300446889046117978, 13.33854408255827387586920177614, 14.47524515780867517424241644882, 17.08849976426933116417692103830

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