↑

Properties

Label 2-112-112.109-c1-0-0
Degree $2$
Conductor $112$
Sign $-0.999 - 0.00414i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.366 + 1.36i)2-s + (−1.86 + 0.5i)3-s + (−1.73 + i)4-s + (−3.23 − 0.866i)5-s + (−1.36 − 2.36i)6-s + (1.73 + 2i)7-s + (−2 − 1.99i)8-s + (0.633 − 0.366i)9-s − 4.73i·10-s + (1.13 + 4.23i)11-s + (2.73 − 2.73i)12-s + (0.267 + 0.267i)13-s + (−2.09 + 3.09i)14-s + 6.46·15-s + (1.99 − 3.46i)16-s + (0.232 − 0.401i)17-s + ⋯
L(s)  = 1  + (0.258 + 0.965i)2-s + (−1.07 + 0.288i)3-s + (−0.866 + 0.5i)4-s + (−1.44 − 0.387i)5-s + (−0.557 − 0.965i)6-s + (0.654 + 0.755i)7-s + (−0.707 − 0.707i)8-s + (0.211 − 0.122i)9-s − 1.49i·10-s + (0.341 + 1.27i)11-s + (0.788 − 0.788i)12-s + (0.0743 + 0.0743i)13-s + (−0.560 + 0.827i)14-s + 1.66·15-s + (0.499 − 0.866i)16-s + (0.0562 − 0.0974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00414i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00414i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.999 - 0.00414i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ -0.999 - 0.00414i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00100309 + 0.483498i\)
\(L(\frac12)\) \(\approx\) \(0.00100309 + 0.483498i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.366 - 1.36i)T \)
7 \( 1 + (-1.73 - 2i)T \)
good3 \( 1 + (1.86 - 0.5i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (3.23 + 0.866i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.13 - 4.23i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-0.267 - 0.267i)T + 13iT^{2} \)
17 \( 1 + (-0.232 + 0.401i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.13 - 4.23i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (2.13 - 1.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.73 + 3.73i)T + 29iT^{2} \)
31 \( 1 + (0.133 - 0.232i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-10.6 - 2.86i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 8.92iT - 41T^{2} \)
43 \( 1 + (0.464 - 0.464i)T - 43iT^{2} \)
47 \( 1 + (3.86 + 6.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.96 + 11.0i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-2.66 - 9.96i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.0358 - 0.133i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (7.33 - 1.96i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.46iT - 71T^{2} \)
73 \( 1 + (2.76 + 1.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.330 + 0.571i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.46 - 8.46i)T + 83iT^{2} \)
89 \( 1 + (-4.5 + 2.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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

−14.69427157142081834665404007382, −12.89630055210795965492628580023, −11.89586084031937860641267603342, −11.58117032717245954832359965229, −9.807423142131706131226735397953, −8.376393599888754671186427207463, −7.60779434664934215152458978027, −6.15391997458222707588993286851, −4.94498367535783787669858701095, −4.15288142507964309005728556221, 0.58588309031003057386554741202, 3.46924708563469755319202440595, 4.63615321022552839673244963025, 6.11451092467452462595755788918, 7.60005587078728445509994086171, 8.822685100248811415874065165939, 10.77169302600324448034758598834, 11.13106901208710172496590654615, 11.72979174606804594277719685038, 12.76322620464809061397078967414

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