↑

Properties

Label 2-112-28.3-c5-0-15
Degree $2$
Conductor $112$
Sign $0.222 + 0.974i$
Analytic cond. $17.9629$
Root an. cond. $4.23827$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−4.77 + 8.26i)3-s + (88.0 − 50.8i)5-s + (−12.6 − 129. i)7-s + (75.9 + 131. i)9-s + (−496. − 286. i)11-s + 25.1i·13-s + 971. i·15-s + (−1.12e3 − 651. i)17-s + (−425. − 736. i)19-s + (1.12e3 + 511. i)21-s + (1.90e3 − 1.09e3i)23-s + (3.61e3 − 6.25e3i)25-s − 3.76e3·27-s + 7.13e3·29-s + (1.63e3 − 2.82e3i)31-s + ⋯
L(s)  = 1  + (−0.306 + 0.530i)3-s + (1.57 − 0.909i)5-s + (−0.0975 − 0.995i)7-s + (0.312 + 0.541i)9-s + (−1.23 − 0.714i)11-s + 0.0412i·13-s + 1.11i·15-s + (−0.947 − 0.547i)17-s + (−0.270 − 0.468i)19-s + (0.557 + 0.253i)21-s + (0.750 − 0.433i)23-s + (1.15 − 2.00i)25-s − 0.995·27-s + 1.57·29-s + (0.305 − 0.528i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.222 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.222 + 0.974i$
Analytic conductor: \(17.9629\)
Root analytic conductor: \(4.23827\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :5/2),\ 0.222 + 0.974i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.803200051\)
\(L(\frac12)\) \(\approx\) \(1.803200051\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (12.6 + 129. i)T \)
good3 \( 1 + (4.77 - 8.26i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-88.0 + 50.8i)T + (1.56e3 - 2.70e3i)T^{2} \)
11 \( 1 + (496. + 286. i)T + (8.05e4 + 1.39e5i)T^{2} \)
13 \( 1 - 25.1iT - 3.71e5T^{2} \)
17 \( 1 + (1.12e3 + 651. i)T + (7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (425. + 736. i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-1.90e3 + 1.09e3i)T + (3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 - 7.13e3T + 2.05e7T^{2} \)
31 \( 1 + (-1.63e3 + 2.82e3i)T + (-1.43e7 - 2.47e7i)T^{2} \)
37 \( 1 + (6.76e3 + 1.17e4i)T + (-3.46e7 + 6.00e7i)T^{2} \)
41 \( 1 - 169. iT - 1.15e8T^{2} \)
43 \( 1 - 1.74e4iT - 1.47e8T^{2} \)
47 \( 1 + (-325. - 563. i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-4.90e3 + 8.49e3i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (-2.07e4 + 3.58e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (-1.00e4 + 5.82e3i)T + (4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-2.65e4 - 1.53e4i)T + (6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 - 3.27e4iT - 1.80e9T^{2} \)
73 \( 1 + (1.80e4 + 1.03e4i)T + (1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (1.11e4 - 6.43e3i)T + (1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 - 1.20e5T + 3.93e9T^{2} \)
89 \( 1 + (8.64e4 - 4.98e4i)T + (2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 - 4.67e4iT - 8.58e9T^{2} \)
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

−12.95346242401100272851881570758, −11.00001148150597330723767196450, −10.35607311429702723725447339046, −9.460243408537603245945468343089, −8.268591566408404023733522227746, −6.66361790892690376137537887757, −5.30822294694136744185764502765, −4.59949268831844979797008630419, −2.39850028643736890337937069381, −0.68282864356705516945953942753, 1.77873410526744985801004000151, 2.78433604026216446616848619881, 5.21191076256323090419709048503, 6.24515477654971301779895935443, 6.99072561222610964095871200702, 8.706035359725092209967282383843, 9.916247831445332720768921272124, 10.58105197933733973075821678172, 12.06463098224455897887953980372, 12.96221665460159338979041418263

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