↑

Properties

Label 2-12e2-144.85-c1-0-19
Degree $2$
Conductor $144$
Sign $-0.460 + 0.887i$
Analytic cond. $1.14984$
Root an. cond. $1.07230$
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  + (0.902 − 1.08i)2-s + (−1.15 − 1.28i)3-s + (−0.369 − 1.96i)4-s + (1.21 − 0.326i)5-s + (−2.44 + 0.0946i)6-s + (−0.707 + 0.408i)7-s + (−2.47 − 1.37i)8-s + (−0.324 + 2.98i)9-s + (0.743 − 1.61i)10-s + (0.497 − 1.85i)11-s + (−2.10 + 2.74i)12-s + (0.116 + 0.434i)13-s + (−0.194 + 1.13i)14-s + (−1.82 − 1.19i)15-s + (−3.72 + 1.45i)16-s + 6.62·17-s + ⋯
L(s)  = 1  + (0.638 − 0.769i)2-s + (−0.667 − 0.744i)3-s + (−0.184 − 0.982i)4-s + (0.544 − 0.145i)5-s + (−0.999 + 0.0386i)6-s + (−0.267 + 0.154i)7-s + (−0.874 − 0.485i)8-s + (−0.108 + 0.994i)9-s + (0.235 − 0.511i)10-s + (0.149 − 0.559i)11-s + (−0.608 + 0.793i)12-s + (0.0322 + 0.120i)13-s + (−0.0519 + 0.304i)14-s + (−0.471 − 0.307i)15-s + (−0.931 + 0.363i)16-s + 1.60·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(144\)    =    \(2^{4} \cdot 3^{2}\)
Sign: $-0.460 + 0.887i$
Analytic conductor: \(1.14984\)
Root analytic conductor: \(1.07230\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{144} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 144,\ (\ :1/2),\ -0.460 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656960 - 1.08105i\)
\(L(\frac12)\) \(\approx\) \(0.656960 - 1.08105i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.902 + 1.08i)T \)
3 \( 1 + (1.15 + 1.28i)T \)
good5 \( 1 + (-1.21 + 0.326i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (0.707 - 0.408i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.497 + 1.85i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.116 - 0.434i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 - 6.62T + 17T^{2} \)
19 \( 1 + (-1.18 + 1.18i)T - 19iT^{2} \)
23 \( 1 + (-2.66 - 1.53i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.65 - 2.31i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (4.61 - 7.99i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.14 + 2.14i)T + 37iT^{2} \)
41 \( 1 + (9.15 + 5.28i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.66 + 6.19i)T + (-37.2 - 21.5i)T^{2} \)
47 \( 1 + (0.140 + 0.244i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.83 - 4.83i)T + 53iT^{2} \)
59 \( 1 + (7.15 - 1.91i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.87 + 2.64i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.39 - 5.22i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.27iT - 71T^{2} \)
73 \( 1 + 4.92iT - 73T^{2} \)
79 \( 1 + (-7.70 - 13.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (10.9 + 2.92i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + 3.44iT - 89T^{2} \)
97 \( 1 + (4.46 + 7.74i)T + (-48.5 + 84.0i)T^{2} \)
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

−12.52423851139732351962607702303, −12.06978537657750845769286968760, −10.92277303333903335584103883095, −10.05349993974898792994374534415, −8.787917388201593722213459984263, −7.08017662001363648787905172569, −5.87356411247727092162613814452, −5.14839540283356301146356608921, −3.17414677051276182582142326782, −1.38891871242369128678628774449, 3.31681388712031667914987407403, 4.66790464191602733640124911210, 5.74689849400389283548912612405, 6.62582180817778408026905690278, 7.982589610127374524587763357603, 9.494486659082235759529409764404, 10.21067890159623935451631083330, 11.68358656175764093180582284013, 12.42074222845802291907537653755, 13.56742313585908492676327506713

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