↑

Properties

Label 2-126-63.13-c2-0-12
Degree $2$
Conductor $126$
Sign $-0.693 + 0.720i$
Analytic cond. $3.43325$
Root an. cond. $1.85290$
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  + (−0.707 − 1.22i)2-s + (−2.47 + 1.69i)3-s + (−0.999 + 1.73i)4-s + (2.08 + 1.20i)5-s + (3.82 + 1.83i)6-s + (−3.20 − 6.22i)7-s + 2.82·8-s + (3.25 − 8.38i)9-s − 3.40i·10-s + (−6.79 − 11.7i)11-s + (−0.458 − 5.98i)12-s + (−11.7 − 6.78i)13-s + (−5.34 + 8.32i)14-s + (−7.19 + 0.551i)15-s + (−2.00 − 3.46i)16-s + 19.8i·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.825 + 0.564i)3-s + (−0.249 + 0.433i)4-s + (0.416 + 0.240i)5-s + (0.637 + 0.305i)6-s + (−0.458 − 0.888i)7-s + 0.353·8-s + (0.362 − 0.932i)9-s − 0.340i·10-s + (−0.618 − 1.07i)11-s + (−0.0382 − 0.498i)12-s + (−0.904 − 0.522i)13-s + (−0.382 + 0.594i)14-s + (−0.479 + 0.0367i)15-s + (−0.125 − 0.216i)16-s + 1.16i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(126\)    =    \(2 \cdot 3^{2} \cdot 7\)
Sign: $-0.693 + 0.720i$
Analytic conductor: \(3.43325\)
Root analytic conductor: \(1.85290\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{126} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 126,\ (\ :1),\ -0.693 + 0.720i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.202674 - 0.476777i\)
\(L(\frac12)\) \(\approx\) \(0.202674 - 0.476777i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 + (2.47 - 1.69i)T \)
7 \( 1 + (3.20 + 6.22i)T \)
good5 \( 1 + (-2.08 - 1.20i)T + (12.5 + 21.6i)T^{2} \)
11 \( 1 + (6.79 + 11.7i)T + (-60.5 + 104. i)T^{2} \)
13 \( 1 + (11.7 + 6.78i)T + (84.5 + 146. i)T^{2} \)
17 \( 1 - 19.8iT - 289T^{2} \)
19 \( 1 + 34.4iT - 361T^{2} \)
23 \( 1 + (-4.33 + 7.50i)T + (-264.5 - 458. i)T^{2} \)
29 \( 1 + (-12.9 - 22.4i)T + (-420.5 + 728. i)T^{2} \)
31 \( 1 + (21.6 + 12.5i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 - 39.6T + 1.36e3T^{2} \)
41 \( 1 + (41.8 + 24.1i)T + (840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-13.1 - 22.7i)T + (-924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (14.5 - 8.41i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 - 35.6T + 2.80e3T^{2} \)
59 \( 1 + (-63.9 - 36.9i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (78.5 - 45.3i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (10.3 - 17.9i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 38.1T + 5.04e3T^{2} \)
73 \( 1 - 12.0iT - 5.32e3T^{2} \)
79 \( 1 + (65.4 + 113. i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (27.6 - 15.9i)T + (3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 118. iT - 7.92e3T^{2} \)
97 \( 1 + (-99.4 + 57.4i)T + (4.70e3 - 8.14e3i)T^{2} \)
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

−12.73219640248723223104921170222, −11.38918304734571152812679678711, −10.55901662466256912101013057826, −10.06266600768669020117688253272, −8.801140625198475780787955887276, −7.23169178933301016471040893345, −5.95313103545249842723329529043, −4.53168540587370411256269993834, −3.03966564174970607842717192997, −0.42841328130935878991117745061, 2.03632962797165616626940255364, 4.93824571197928411647554480000, 5.76786505664765386184296269066, 6.95889616130410204446200595566, 7.897105833695326295801577498805, 9.479051676289826128663760956073, 10.08154659145933472077732448112, 11.70926099353254477622768487646, 12.44867969086112686600608952342, 13.38927907529466459141884698157

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