Properties

Label 2-2e2-1.1-c5-0-0
Degree $2$
Conductor $4$
Sign $1$
Analytic cond. $0.641535$
Root an. cond. $0.800958$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 12·3-s + 54·5-s − 88·7-s − 99·9-s + 540·11-s − 418·13-s − 648·15-s + 594·17-s + 836·19-s + 1.05e3·21-s − 4.10e3·23-s − 209·25-s + 4.10e3·27-s − 594·29-s + 4.25e3·31-s − 6.48e3·33-s − 4.75e3·35-s − 298·37-s + 5.01e3·39-s + 1.72e4·41-s − 1.21e4·43-s − 5.34e3·45-s − 1.29e3·47-s − 9.06e3·49-s − 7.12e3·51-s + 1.94e4·53-s + 2.91e4·55-s + ⋯
L(s)  = 1  − 0.769·3-s + 0.965·5-s − 0.678·7-s − 0.407·9-s + 1.34·11-s − 0.685·13-s − 0.743·15-s + 0.498·17-s + 0.531·19-s + 0.522·21-s − 1.61·23-s − 0.0668·25-s + 1.08·27-s − 0.131·29-s + 0.795·31-s − 1.03·33-s − 0.655·35-s − 0.0357·37-s + 0.528·39-s + 1.60·41-s − 0.997·43-s − 0.393·45-s − 0.0855·47-s − 0.539·49-s − 0.383·51-s + 0.953·53-s + 1.29·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $1$
Analytic conductor: \(0.641535\)
Root analytic conductor: \(0.800958\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{4} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7805376825\)
\(L(\frac12)\) \(\approx\) \(0.7805376825\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + 4 p T + p^{5} T^{2} \)
5 \( 1 - 54 T + p^{5} T^{2} \)
7 \( 1 + 88 T + p^{5} T^{2} \)
11 \( 1 - 540 T + p^{5} T^{2} \)
13 \( 1 + 418 T + p^{5} T^{2} \)
17 \( 1 - 594 T + p^{5} T^{2} \)
19 \( 1 - 44 p T + p^{5} T^{2} \)
23 \( 1 + 4104 T + p^{5} T^{2} \)
29 \( 1 + 594 T + p^{5} T^{2} \)
31 \( 1 - 4256 T + p^{5} T^{2} \)
37 \( 1 + 298 T + p^{5} T^{2} \)
41 \( 1 - 17226 T + p^{5} T^{2} \)
43 \( 1 + 12100 T + p^{5} T^{2} \)
47 \( 1 + 1296 T + p^{5} T^{2} \)
53 \( 1 - 19494 T + p^{5} T^{2} \)
59 \( 1 + 7668 T + p^{5} T^{2} \)
61 \( 1 + 34738 T + p^{5} T^{2} \)
67 \( 1 - 21812 T + p^{5} T^{2} \)
71 \( 1 + 46872 T + p^{5} T^{2} \)
73 \( 1 - 67562 T + p^{5} T^{2} \)
79 \( 1 + 76912 T + p^{5} T^{2} \)
83 \( 1 - 67716 T + p^{5} T^{2} \)
89 \( 1 - 29754 T + p^{5} T^{2} \)
97 \( 1 + 122398 T + p^{5} T^{2} \)
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

−24.66474956351914942067996001362, −22.76638104045062086910631624859, −21.82648947337359762085007791626, −19.75976565483922320580773419775, −17.71694741483574255598828058356, −16.59946943253973234810046665847, −14.11508352828821016305212270051, −11.99964135123060776621941064917, −9.747199643115818989680697413553, −6.10006048288389002479771045385, 6.10006048288389002479771045385, 9.747199643115818989680697413553, 11.99964135123060776621941064917, 14.11508352828821016305212270051, 16.59946943253973234810046665847, 17.71694741483574255598828058356, 19.75976565483922320580773419775, 21.82648947337359762085007791626, 22.76638104045062086910631624859, 24.66474956351914942067996001362

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