↑

Properties

Label 2-156-156.35-c1-0-6
Degree $2$
Conductor $156$
Sign $-0.306 - 0.951i$
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  + (0.998 + 1.00i)2-s + (−0.605 + 1.62i)3-s + (−0.00424 + 1.99i)4-s − 0.171i·5-s + (−2.22 + 1.01i)6-s + (2.51 − 1.45i)7-s + (−2.00 + 1.99i)8-s + (−2.26 − 1.96i)9-s + (0.171 − 0.171i)10-s + (−0.674 + 1.16i)11-s + (−3.24 − 1.21i)12-s + (−3.29 + 1.45i)13-s + (3.97 + 1.06i)14-s + (0.278 + 0.103i)15-s + (−3.99 − 0.0169i)16-s + (4.48 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.706 + 0.707i)2-s + (−0.349 + 0.936i)3-s + (−0.00212 + 0.999i)4-s − 0.0767i·5-s + (−0.910 + 0.414i)6-s + (0.952 − 0.549i)7-s + (−0.709 + 0.704i)8-s + (−0.755 − 0.654i)9-s + (0.0543 − 0.0541i)10-s + (−0.203 + 0.352i)11-s + (−0.936 − 0.351i)12-s + (−0.915 + 0.403i)13-s + (1.06 + 0.285i)14-s + (0.0718 + 0.0268i)15-s + (−0.999 − 0.00424i)16-s + (1.08 − 0.628i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.306 - 0.951i$
Analytic conductor: \(1.24566\)
Root analytic conductor: \(1.11609\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1/2),\ -0.306 - 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.857559 + 1.17679i\)
\(L(\frac12)\) \(\approx\) \(0.857559 + 1.17679i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.998 - 1.00i)T \)
3 \( 1 + (0.605 - 1.62i)T \)
13 \( 1 + (3.29 - 1.45i)T \)
good5 \( 1 + 0.171iT - 5T^{2} \)
7 \( 1 + (-2.51 + 1.45i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.674 - 1.16i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-4.48 + 2.59i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.90 + 2.82i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.40 + 2.44i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.51 + 1.45i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.62iT - 31T^{2} \)
37 \( 1 + (1.64 - 2.85i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.85 + 2.80i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.79 - 3.92i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 7.05T + 47T^{2} \)
53 \( 1 + 12.1iT - 53T^{2} \)
59 \( 1 + (-4.93 - 8.55i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.259 + 0.449i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.87 + 5.70i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (5.79 + 10.0i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 1.09T + 73T^{2} \)
79 \( 1 + 5.99iT - 79T^{2} \)
83 \( 1 - 3.19T + 83T^{2} \)
89 \( 1 + (-3.87 - 2.23i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.06 + 7.03i)T + (-48.5 + 84.0i)T^{2} \)
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

−13.49250710065626685080046739180, −12.09006096536857838419071712508, −11.50385590465746744553018093399, −10.24443886028015167999060542505, −9.103675028189436145597944811581, −7.82888743541136071329595295282, −6.82367616879017494190580891608, −5.07320570719844894358772591243, −4.83147903673103981449008048290, −3.21868339439269603445367691856, 1.54866222805574192835962944361, 3.07011103497372060138822302969, 5.13208153645232546012354306316, 5.71895967816473012959995120989, 7.27073735396350240597651518708, 8.366764554396340686265174261489, 9.908577538536051463262109470569, 11.02573803419200003884444280408, 11.85622260805605169665685913055, 12.46096693258718623262306744087

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