↑

Properties

Label 2-112-112.109-c1-0-11
Degree $2$
Conductor $112$
Sign $0.496 + 0.868i$
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  + (1 − i)2-s + (0.5 − 0.133i)3-s − 2i·4-s + (−0.866 − 0.232i)5-s + (0.366 − 0.633i)6-s + (1.73 + 2i)7-s + (−2 − 2i)8-s + (−2.36 + 1.36i)9-s + (−1.09 + 0.633i)10-s + (−0.767 − 2.86i)11-s + (−0.267 − i)12-s + (3.73 + 3.73i)13-s + (3.73 + 0.267i)14-s − 0.464·15-s − 4·16-s + (−3.23 + 5.59i)17-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (0.288 − 0.0773i)3-s − i·4-s + (−0.387 − 0.103i)5-s + (0.149 − 0.258i)6-s + (0.654 + 0.755i)7-s + (−0.707 − 0.707i)8-s + (−0.788 + 0.455i)9-s + (−0.347 + 0.200i)10-s + (−0.231 − 0.864i)11-s + (−0.0773 − 0.288i)12-s + (1.03 + 1.03i)13-s + (0.997 + 0.0716i)14-s − 0.119·15-s − 16-s + (−0.783 + 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.496 + 0.868i$
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: \((2,\ 112,\ (\ :1/2),\ 0.496 + 0.868i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26768 - 0.735407i\)
\(L(\frac12)\) \(\approx\) \(1.26768 - 0.735407i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 + i)T \)
7 \( 1 + (-1.73 - 2i)T \)
good3 \( 1 + (-0.5 + 0.133i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (0.866 + 0.232i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.767 + 2.86i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.73 - 3.73i)T + 13iT^{2} \)
17 \( 1 + (3.23 - 5.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.767 + 2.86i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.86 + 2.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.267 + 0.267i)T + 29iT^{2} \)
31 \( 1 + (1.86 - 3.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.13 + 0.303i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.92iT - 41T^{2} \)
43 \( 1 + (-6.46 + 6.46i)T - 43iT^{2} \)
47 \( 1 + (2.13 + 3.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.06 + 3.96i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.03 + 11.3i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.86 + 6.96i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (4.96 - 1.33i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.535iT - 71T^{2} \)
73 \( 1 + (-6.23 - 3.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.33 - 14.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.53 - 1.53i)T + 83iT^{2} \)
89 \( 1 + (4.5 - 2.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.92T + 97T^{2} \)
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

−13.54462965261927892050200324560, −12.39751543158063897097847014161, −11.18822369564559784732056072355, −10.98548764473402107256961460889, −8.972245469777042854274095188035, −8.402782309814765828765097737070, −6.36897127787173522287939823319, −5.21586234007550947967474283453, −3.76213617381219925814576176267, −2.15882881959856427094114440845, 3.15932286533652217046788705721, 4.45794465664610703107600371842, 5.79298795266145422142016823081, 7.29849330920029166029439475402, 8.015743297465138467801449120467, 9.265692886026722219942833852813, 10.98855322002277922431199774985, 11.78580366198680379087343270790, 13.11933758438720392988526992277, 13.85470882898036615365622873506

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