↑

Properties

Label 2-6e2-36.7-c2-0-1
Degree $2$
Conductor $36$
Sign $0.716 - 0.697i$
Analytic cond. $0.980928$
Root an. cond. $0.990418$
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  + (−1.84 − 0.778i)2-s + (0.262 + 2.98i)3-s + (2.78 + 2.86i)4-s + (1.10 + 1.90i)5-s + (1.84 − 5.70i)6-s + (7.23 + 4.17i)7-s + (−2.90 − 7.45i)8-s + (−8.86 + 1.56i)9-s + (−0.544 − 4.36i)10-s + (−4.54 − 2.62i)11-s + (−7.83 + 9.08i)12-s + (−7.37 − 12.7i)13-s + (−10.0 − 13.3i)14-s + (−5.40 + 3.79i)15-s + (−0.450 + 15.9i)16-s + 28.2·17-s + ⋯
L(s)  = 1  + (−0.921 − 0.389i)2-s + (0.0874 + 0.996i)3-s + (0.697 + 0.716i)4-s + (0.220 + 0.381i)5-s + (0.307 − 0.951i)6-s + (1.03 + 0.597i)7-s + (−0.363 − 0.931i)8-s + (−0.984 + 0.174i)9-s + (−0.0544 − 0.436i)10-s + (−0.413 − 0.238i)11-s + (−0.653 + 0.757i)12-s + (−0.567 − 0.982i)13-s + (−0.720 − 0.952i)14-s + (−0.360 + 0.252i)15-s + (−0.0281 + 0.999i)16-s + 1.66·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.716 - 0.697i$
Analytic conductor: \(0.980928\)
Root analytic conductor: \(0.990418\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 36,\ (\ :1),\ 0.716 - 0.697i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.718868 + 0.292069i\)
\(L(\frac12)\) \(\approx\) \(0.718868 + 0.292069i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.84 + 0.778i)T \)
3 \( 1 + (-0.262 - 2.98i)T \)
good5 \( 1 + (-1.10 - 1.90i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-7.23 - 4.17i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (4.54 + 2.62i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (7.37 + 12.7i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 28.2T + 289T^{2} \)
19 \( 1 + 19.1iT - 361T^{2} \)
23 \( 1 + (-3.16 + 1.82i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (12.3 - 21.3i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (32.9 - 19.0i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 4.21T + 1.36e3T^{2} \)
41 \( 1 + (9.92 + 17.1i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-20.1 - 11.6i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (-25.8 - 14.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 32.1T + 2.80e3T^{2} \)
59 \( 1 + (7.96 - 4.59i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (40.8 - 70.7i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-6.86 + 3.96i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 + 62.9iT - 5.04e3T^{2} \)
73 \( 1 - 33.3T + 5.32e3T^{2} \)
79 \( 1 + (-53.7 - 31.0i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (103. + 59.4i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 107.T + 7.92e3T^{2} \)
97 \( 1 + (-1.78 + 3.09i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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

−16.52329510001328060298877239737, −15.34051460412875832401570971033, −14.43064101864359474582287883884, −12.36241353906624748829370896814, −11.05727645899445669962651858582, −10.25233361595376134142736602345, −8.931613907031461103209385235015, −7.74785473133191046343252926790, −5.35821200427815016488879171319, −2.92130820260622556641288508592, 1.60467221093575832094185885641, 5.52251931447887224117902547907, 7.30192365069932613183918353523, 8.067702567015005797957709078342, 9.593395293120601538986957743650, 11.15603700248299012162763328764, 12.33147024189918533821307606370, 14.02559677949256955238080023166, 14.75748518858765530110351483124, 16.73730161224291670777808653903

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