↑

Properties

Label 2-156-52.51-c2-0-22
Degree $2$
Conductor $156$
Sign $-0.979 + 0.199i$
Analytic cond. $4.25069$
Root an. cond. $2.06172$
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  + (−1.99 + 0.143i)2-s − 1.73i·3-s + (3.95 − 0.572i)4-s − 8.96i·5-s + (0.248 + 3.45i)6-s − 7.15·7-s + (−7.81 + 1.71i)8-s − 2.99·9-s + (1.28 + 17.8i)10-s + 9.63·11-s + (−0.991 − 6.85i)12-s + (0.746 + 12.9i)13-s + (14.2 − 1.02i)14-s − 15.5·15-s + (15.3 − 4.53i)16-s − 18.0·17-s + ⋯
L(s)  = 1  + (−0.997 + 0.0717i)2-s − 0.577i·3-s + (0.989 − 0.143i)4-s − 1.79i·5-s + (0.0414 + 0.575i)6-s − 1.02·7-s + (−0.976 + 0.213i)8-s − 0.333·9-s + (0.128 + 1.78i)10-s + 0.876·11-s + (−0.0826 − 0.571i)12-s + (0.0574 + 0.998i)13-s + (1.01 − 0.0733i)14-s − 1.03·15-s + (0.959 − 0.283i)16-s − 1.06·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.979 + 0.199i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $-0.979 + 0.199i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ -0.979 + 0.199i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0529091 - 0.524338i\)
\(L(\frac12)\) \(\approx\) \(0.0529091 - 0.524338i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.99 - 0.143i)T \)
3 \( 1 + 1.73iT \)
13 \( 1 + (-0.746 - 12.9i)T \)
good5 \( 1 + 8.96iT - 25T^{2} \)
7 \( 1 + 7.15T + 49T^{2} \)
11 \( 1 - 9.63T + 121T^{2} \)
17 \( 1 + 18.0T + 289T^{2} \)
19 \( 1 + 23.1T + 361T^{2} \)
23 \( 1 + 25.4iT - 529T^{2} \)
29 \( 1 + 2.34T + 841T^{2} \)
31 \( 1 - 40.4T + 961T^{2} \)
37 \( 1 + 20.7iT - 1.36e3T^{2} \)
41 \( 1 + 43.4iT - 1.68e3T^{2} \)
43 \( 1 + 20.0iT - 1.84e3T^{2} \)
47 \( 1 - 19.7T + 2.20e3T^{2} \)
53 \( 1 + 55.3T + 2.80e3T^{2} \)
59 \( 1 + 45.1T + 3.48e3T^{2} \)
61 \( 1 - 106.T + 3.72e3T^{2} \)
67 \( 1 - 62.9T + 4.48e3T^{2} \)
71 \( 1 - 4.89T + 5.04e3T^{2} \)
73 \( 1 - 70.7iT - 5.32e3T^{2} \)
79 \( 1 + 86.8iT - 6.24e3T^{2} \)
83 \( 1 + 68.7T + 6.88e3T^{2} \)
89 \( 1 + 64.1iT - 7.92e3T^{2} \)
97 \( 1 + 118. iT - 9.40e3T^{2} \)
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

−12.34625986987376894050504628220, −11.32592923257334022797199293435, −9.825640415044889305294050707046, −8.847751291869425126896681467170, −8.553737507552991724987597115381, −6.87925037910215504857082370117, −6.12466301831082297912786772487, −4.31328159364342156419331675066, −1.98073463191256716660789161212, −0.43375792475421206358538970675, 2.65158983865884880054531626172, 3.61582981237980548187962956345, 6.27548350637670257942674167636, 6.67023150462633251091607024947, 8.039265837608781248847700588602, 9.403259531649217506893909762066, 10.12883665703414142121072331179, 10.88772065935331887463877111593, 11.65189104607436014931204844186, 13.11968206987292573042275010676

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