Properties

Label 2-648-648.155-c1-0-43
Degree $2$
Conductor $648$
Sign $-0.459 - 0.888i$
Analytic cond. $5.17430$
Root an. cond. $2.27471$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.940 + 1.05i)2-s + (1.70 − 0.275i)3-s + (−0.229 + 1.98i)4-s + (−2.15 + 1.41i)5-s + (1.90 + 1.54i)6-s + (1.15 + 1.09i)7-s + (−2.31 + 1.62i)8-s + (2.84 − 0.943i)9-s + (−3.51 − 0.940i)10-s + (0.0977 + 0.194i)11-s + (0.155 + 3.46i)12-s + (−2.10 + 0.906i)13-s + (−0.0638 + 2.24i)14-s + (−3.28 + 3.01i)15-s + (−3.89 − 0.911i)16-s + (3.39 + 4.04i)17-s + ⋯
L(s)  = 1  + (0.665 + 0.746i)2-s + (0.987 − 0.159i)3-s + (−0.114 + 0.993i)4-s + (−0.962 + 0.633i)5-s + (0.775 + 0.631i)6-s + (0.436 + 0.412i)7-s + (−0.817 + 0.575i)8-s + (0.949 − 0.314i)9-s + (−1.11 − 0.297i)10-s + (0.0294 + 0.0587i)11-s + (0.0449 + 0.998i)12-s + (−0.582 + 0.251i)13-s + (−0.0170 + 0.600i)14-s + (−0.849 + 0.778i)15-s + (−0.973 − 0.227i)16-s + (0.823 + 0.981i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.459 - 0.888i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ -0.459 - 0.888i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27149 + 2.08887i\)
\(L(\frac12)\) \(\approx\) \(1.27149 + 2.08887i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.940 - 1.05i)T \)
3 \( 1 + (-1.70 + 0.275i)T \)
good5 \( 1 + (2.15 - 1.41i)T + (1.98 - 4.59i)T^{2} \)
7 \( 1 + (-1.15 - 1.09i)T + (0.407 + 6.98i)T^{2} \)
11 \( 1 + (-0.0977 - 0.194i)T + (-6.56 + 8.82i)T^{2} \)
13 \( 1 + (2.10 - 0.906i)T + (8.92 - 9.45i)T^{2} \)
17 \( 1 + (-3.39 - 4.04i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (2.51 + 2.11i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (-2.30 - 2.44i)T + (-1.33 + 22.9i)T^{2} \)
29 \( 1 + (-5.78 + 0.676i)T + (28.2 - 6.68i)T^{2} \)
31 \( 1 + (0.779 + 3.28i)T + (-27.7 + 13.9i)T^{2} \)
37 \( 1 + (6.62 - 1.16i)T + (34.7 - 12.6i)T^{2} \)
41 \( 1 + (-3.98 - 2.96i)T + (11.7 + 39.2i)T^{2} \)
43 \( 1 + (-0.594 + 10.2i)T + (-42.7 - 4.99i)T^{2} \)
47 \( 1 + (-0.899 - 0.213i)T + (42.0 + 21.0i)T^{2} \)
53 \( 1 + (0.396 + 0.686i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.32 - 2.64i)T + (-35.2 - 47.3i)T^{2} \)
61 \( 1 + (-12.8 + 3.85i)T + (50.9 - 33.5i)T^{2} \)
67 \( 1 + (-12.6 - 1.47i)T + (65.1 + 15.4i)T^{2} \)
71 \( 1 + (-5.56 - 2.02i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (5.00 - 1.82i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-8.95 + 6.66i)T + (22.6 - 75.6i)T^{2} \)
83 \( 1 + (7.14 - 5.31i)T + (23.8 - 79.5i)T^{2} \)
89 \( 1 + (-1.19 - 3.28i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (4.49 + 2.95i)T + (38.4 + 89.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.00385162883878185917412465747, −9.791365961019300922566478795687, −8.628199372487289667858618192990, −8.138284643292828131331607980257, −7.29347754176355258978614793275, −6.68677332196815764373950105768, −5.30974333777119223443734981588, −4.16022610114552222843312932699, −3.42623858437218419020757145460, −2.31517763341585553772528276393, 1.03619972494042857732036146991, 2.59161234297693868887627269480, 3.62738830823986168647785280077, 4.49785581729006325411193688215, 5.14949029214476929654843944580, 6.83484904313343911522046883071, 7.82493740846256682562828289026, 8.562069506024140810141172461667, 9.520058573658212516005630512243, 10.33066581471105571192582827712

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