↑

Properties

Label 2-156-12.11-c1-0-13
Degree $2$
Conductor $156$
Sign $0.417 + 0.908i$
Analytic cond. $1.24566$
Root an. cond. $1.11609$
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  + (−1.11 − 0.866i)2-s + (1.70 − 0.306i)3-s + (0.500 + 1.93i)4-s − 2.09i·5-s + (−2.17 − 1.13i)6-s + 0.613i·7-s + (1.11 − 2.59i)8-s + (2.81 − 1.04i)9-s + (−1.81 + 2.33i)10-s + (1.44 + 3.14i)12-s + 13-s + (0.531 − 0.686i)14-s + (−0.641 − 3.56i)15-s + (−3.5 + 1.93i)16-s + 2.09i·17-s + (−4.04 − 1.26i)18-s + ⋯
L(s)  = 1  + (−0.790 − 0.612i)2-s + (0.984 − 0.177i)3-s + (0.250 + 0.968i)4-s − 0.935i·5-s + (−0.886 − 0.462i)6-s + 0.231i·7-s + (0.395 − 0.918i)8-s + (0.937 − 0.348i)9-s + (−0.572 + 0.739i)10-s + (0.417 + 0.908i)12-s + 0.277·13-s + (0.142 − 0.183i)14-s + (−0.165 − 0.920i)15-s + (−0.875 + 0.484i)16-s + 0.507i·17-s + (−0.954 − 0.298i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.417 + 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.417 + 0.908i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ 0.417 + 0.908i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.898169 - 0.575724i\)
\(L(\frac12)\) \(\approx\) \(0.898169 - 0.575724i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.11 + 0.866i)T \)
3 \( 1 + (-1.70 + 0.306i)T \)
13 \( 1 - T \)
good5 \( 1 + 2.09iT - 5T^{2} \)
7 \( 1 - 0.613iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 - 2.09iT - 17T^{2} \)
19 \( 1 + 5.29iT - 19T^{2} \)
23 \( 1 + 6.81T + 23T^{2} \)
29 \( 1 - 6.92iT - 29T^{2} \)
31 \( 1 - 5.29iT - 31T^{2} \)
37 \( 1 + 5.62T + 37T^{2} \)
41 \( 1 - 11.1iT - 41T^{2} \)
43 \( 1 - 11.1iT - 43T^{2} \)
47 \( 1 + 3.40T + 47T^{2} \)
53 \( 1 + 11.1iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 5.29iT - 67T^{2} \)
71 \( 1 + 3.40T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 + 6.51iT - 79T^{2} \)
83 \( 1 - 13.6T + 83T^{2} \)
89 \( 1 + 2.74iT - 89T^{2} \)
97 \( 1 + 5.24T + 97T^{2} \)
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

−12.76247590744069376185522783374, −11.86663565673852971356491951120, −10.54431261237114090935999956095, −9.449227739613737767024573117678, −8.707058611912045298565740467227, −8.033983999884523542465872988676, −6.72177548097214016342181970540, −4.59374183235862258100938171323, −3.15550880332309876798858489781, −1.55972770750412420638039260477, 2.22317239066231341118762626536, 3.93727542248255481804512273053, 5.86186181935852707966131976504, 7.12640616153609286378873430152, 7.87499901376151318959768612173, 8.927793488187041543519465444116, 10.07177806289127950495129378163, 10.54736388187699643719279342177, 11.96848152711580845856055102844, 13.76620463784534113776377168119

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