↑

Properties

Label 2-2e2-4.3-c44-0-14
Degree $2$
Conductor $4$
Sign $0.881 + 0.472i$
Analytic cond. $49.0478$
Root an. cond. $7.00342$
Motivic weight $44$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (4.06e6 + 1.02e6i)2-s + 4.78e10i·3-s + (1.55e13 + 8.31e12i)4-s − 4.02e15·5-s + (−4.88e16 + 1.94e17i)6-s − 5.21e18i·7-s + (5.45e19 + 4.96e19i)8-s − 1.30e21·9-s + (−1.63e22 − 4.10e21i)10-s − 5.34e22i·11-s + (−3.97e23 + 7.41e23i)12-s + 3.07e24·13-s + (5.32e24 − 2.12e25i)14-s − 1.92e26i·15-s + (1.71e26 + 2.57e26i)16-s − 1.03e27·17-s + ⋯
L(s)  = 1  + (0.969 + 0.243i)2-s + 1.52i·3-s + (0.881 + 0.472i)4-s − 1.68·5-s + (−0.371 + 1.47i)6-s − 1.33i·7-s + (0.739 + 0.672i)8-s − 1.32·9-s + (−1.63 − 0.410i)10-s − 0.656i·11-s + (−0.719 + 1.34i)12-s + 0.956·13-s + (0.324 − 1.29i)14-s − 2.57i·15-s + (0.553 + 0.832i)16-s − 0.885·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(45-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+22) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $0.881 + 0.472i$
Analytic conductor: \(49.0478\)
Root analytic conductor: \(7.00342\)
Motivic weight: \(44\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :22),\ 0.881 + 0.472i)\)

Particular Values

\(L(\frac{45}{2})\) \(\approx\) \(1.762283713\)
\(L(\frac12)\) \(\approx\) \(1.762283713\)
\(L(23)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4.06e6 - 1.02e6i)T \)
good3 \( 1 - 4.78e10iT - 9.84e20T^{2} \)
5 \( 1 + 4.02e15T + 5.68e30T^{2} \)
7 \( 1 + 5.21e18iT - 1.52e37T^{2} \)
11 \( 1 + 5.34e22iT - 6.62e45T^{2} \)
13 \( 1 - 3.07e24T + 1.03e49T^{2} \)
17 \( 1 + 1.03e27T + 1.37e54T^{2} \)
19 \( 1 + 1.75e28iT - 1.84e56T^{2} \)
23 \( 1 + 1.30e29iT - 8.24e59T^{2} \)
29 \( 1 + 1.16e32T + 2.21e64T^{2} \)
31 \( 1 - 2.21e32iT - 4.16e65T^{2} \)
37 \( 1 - 1.46e34T + 1.00e69T^{2} \)
41 \( 1 + 2.56e35T + 9.17e70T^{2} \)
43 \( 1 - 7.73e34iT - 7.45e71T^{2} \)
47 \( 1 + 1.06e37iT - 3.73e73T^{2} \)
53 \( 1 - 5.04e37T + 7.38e75T^{2} \)
59 \( 1 + 1.45e39iT - 8.26e77T^{2} \)
61 \( 1 - 2.53e39T + 3.58e78T^{2} \)
67 \( 1 + 1.88e40iT - 2.22e80T^{2} \)
71 \( 1 + 6.82e40iT - 2.85e81T^{2} \)
73 \( 1 + 5.18e40T + 9.68e81T^{2} \)
79 \( 1 + 1.41e41iT - 3.13e83T^{2} \)
83 \( 1 - 2.71e41iT - 2.75e84T^{2} \)
89 \( 1 + 5.08e42T + 5.93e85T^{2} \)
97 \( 1 + 2.11e43T + 2.61e87T^{2} \)
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

−15.07170672153672844095979623335, −13.48053377120867121856834642847, −11.32411772896041422972284793730, −10.82434914633751778971238699097, −8.454726154844239531821935665692, −6.95379371595648361853236207846, −4.83068417071982694024967430554, −3.94095464020444526449317326390, −3.40289829374845905655311029416, −0.35047397668912534562144092172, 1.31410637660700281384383909691, 2.53497016082375979071352675982, 4.01335976121415520771442683049, 5.86345465688891884619685939598, 7.15335099116746050129664936446, 8.304972327016775437926869800487, 11.40881633678683976565977778410, 12.13404653906992732178133286895, 13.01465382338941740265379463455, 14.83474096729184455901036377053

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