↑

Properties

Label 2-112-7.5-c2-0-1
Degree $2$
Conductor $112$
Sign $0.605 - 0.795i$
Analytic cond. $3.05177$
Root an. cond. $1.74693$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.5 + 0.866i)3-s + (1.5 + 0.866i)5-s + 7·7-s + (−3 + 5.19i)9-s + (7.5 + 12.9i)11-s + 13.8i·13-s − 3·15-s + (25.5 − 14.7i)17-s + (−13.5 − 7.79i)19-s + (−10.5 + 6.06i)21-s + (−4.5 + 7.79i)23-s + (−11 − 19.0i)25-s − 25.9i·27-s − 6·29-s + (10.5 − 6.06i)31-s + ⋯
L(s)  = 1  + (−0.5 + 0.288i)3-s + (0.300 + 0.173i)5-s + 7-s + (−0.333 + 0.577i)9-s + (0.681 + 1.18i)11-s + 1.06i·13-s − 0.200·15-s + (1.5 − 0.866i)17-s + (−0.710 − 0.410i)19-s + (−0.5 + 0.288i)21-s + (−0.195 + 0.338i)23-s + (−0.440 − 0.762i)25-s − 0.962i·27-s − 0.206·29-s + (0.338 − 0.195i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.605 - 0.795i$
Analytic conductor: \(3.05177\)
Root analytic conductor: \(1.74693\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1),\ 0.605 - 0.795i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.16912 + 0.579565i\)
\(L(\frac12)\) \(\approx\) \(1.16912 + 0.579565i\)
\(L(2)\) 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 \( 1 - 7T \)
good3 \( 1 + (1.5 - 0.866i)T + (4.5 - 7.79i)T^{2} \)
5 \( 1 + (-1.5 - 0.866i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (-7.5 - 12.9i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 - 13.8iT - 169T^{2} \)
17 \( 1 + (-25.5 + 14.7i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (13.5 + 7.79i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (4.5 - 7.79i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + 6T + 841T^{2} \)
31 \( 1 + (-10.5 + 6.06i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + (15.5 - 26.8i)T + (-684.5 - 1.18e3i)T^{2} \)
41 \( 1 + 55.4iT - 1.68e3T^{2} \)
43 \( 1 + 10T + 1.84e3T^{2} \)
47 \( 1 + (37.5 + 21.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + (-28.5 - 49.3i)T + (-1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (-70.5 + 40.7i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (70.5 + 40.7i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (24.5 + 42.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 126T + 5.04e3T^{2} \)
73 \( 1 + (22.5 - 12.9i)T + (2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (36.5 - 63.2i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 - 13.8iT - 6.88e3T^{2} \)
89 \( 1 + (-49.5 - 28.5i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 - 27.7iT - 9.40e3T^{2} \)
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

−13.85812791908300495274899568142, −12.12494265750984570131212651931, −11.57517301898252890784134137938, −10.41290321489426471643210572794, −9.439484184654691738400925917574, −8.040030627529234577788485249960, −6.82718351490688586277730332877, −5.35086373085198882901340646873, −4.35102810335025232988051383997, −1.99827174172816095797146914257, 1.18224784871517366181068520123, 3.55227920121510277483970729930, 5.45153242023210553270722603753, 6.17065364639748473024377254691, 7.87601704268262659033521622578, 8.743126392890008527275074788892, 10.23207693595868898588502446850, 11.27308501690876625803415233588, 12.10763757215575182759147965683, 13.11636990209035990742086798502

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