Properties

Label 2-156-4.3-c2-0-6
Degree $2$
Conductor $156$
Sign $0.601 - 0.798i$
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  + (−1.89 − 0.634i)2-s + 1.73i·3-s + (3.19 + 2.40i)4-s + 6.10·5-s + (1.09 − 3.28i)6-s + 5.33i·7-s + (−4.53 − 6.59i)8-s − 2.99·9-s + (−11.5 − 3.87i)10-s + 0.0387i·11-s + (−4.16 + 5.53i)12-s − 3.60·13-s + (3.38 − 10.1i)14-s + 10.5i·15-s + (4.42 + 15.3i)16-s + 19.7·17-s + ⋯
L(s)  = 1  + (−0.948 − 0.317i)2-s + 0.577i·3-s + (0.798 + 0.601i)4-s + 1.22·5-s + (0.183 − 0.547i)6-s + 0.761i·7-s + (−0.566 − 0.823i)8-s − 0.333·9-s + (−1.15 − 0.387i)10-s + 0.00352i·11-s + (−0.347 + 0.461i)12-s − 0.277·13-s + (0.241 − 0.722i)14-s + 0.705i·15-s + (0.276 + 0.961i)16-s + 1.16·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.601 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(156\)    =    \(2^{2} \cdot 3 \cdot 13\)
Sign: $0.601 - 0.798i$
Analytic conductor: \(4.25069\)
Root analytic conductor: \(2.06172\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{156} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 156,\ (\ :1),\ 0.601 - 0.798i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.01378 + 0.505665i\)
\(L(\frac12)\) \(\approx\) \(1.01378 + 0.505665i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.89 + 0.634i)T \)
3 \( 1 - 1.73iT \)
13 \( 1 + 3.60T \)
good5 \( 1 - 6.10T + 25T^{2} \)
7 \( 1 - 5.33iT - 49T^{2} \)
11 \( 1 - 0.0387iT - 121T^{2} \)
17 \( 1 - 19.7T + 289T^{2} \)
19 \( 1 - 29.9iT - 361T^{2} \)
23 \( 1 - 5.79iT - 529T^{2} \)
29 \( 1 - 39.4T + 841T^{2} \)
31 \( 1 - 11.5iT - 961T^{2} \)
37 \( 1 + 14.7T + 1.36e3T^{2} \)
41 \( 1 + 41.9T + 1.68e3T^{2} \)
43 \( 1 - 14.6iT - 1.84e3T^{2} \)
47 \( 1 + 67.1iT - 2.20e3T^{2} \)
53 \( 1 + 69.6T + 2.80e3T^{2} \)
59 \( 1 - 9.23iT - 3.48e3T^{2} \)
61 \( 1 - 69.1T + 3.72e3T^{2} \)
67 \( 1 + 133. iT - 4.48e3T^{2} \)
71 \( 1 - 30.6iT - 5.04e3T^{2} \)
73 \( 1 - 63.3T + 5.32e3T^{2} \)
79 \( 1 + 145. iT - 6.24e3T^{2} \)
83 \( 1 + 118. iT - 6.88e3T^{2} \)
89 \( 1 + 75.1T + 7.92e3T^{2} \)
97 \( 1 + 49.2T + 9.40e3T^{2} \)
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

−12.48838040949114903773833747217, −11.80932988906997899288681800011, −10.29549920761504357402511728875, −9.981132301893041993081989087426, −8.979906166126233515262463921930, −7.984017900708793132576983624209, −6.36400540916595880478352009716, −5.38718029033744403944001948984, −3.26257490640666178637440415687, −1.82064805492591995083353124258, 1.05845871556836820789407350465, 2.59315255921651399640004044913, 5.22154937467733645059705565285, 6.41376456255878479291635933378, 7.22358268557593887457648112496, 8.393237404219145361379737006259, 9.565257015127394965370960806817, 10.25774134138067223189206264862, 11.31506016846058085445946140578, 12.57345516956877714619315663735

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