Properties

Label 2-648-648.155-c1-0-47
Degree $2$
Conductor $648$
Sign $0.937 - 0.347i$
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  + (−0.849 − 1.13i)2-s + (0.956 + 1.44i)3-s + (−0.556 + 1.92i)4-s + (1.24 − 0.819i)5-s + (0.820 − 2.30i)6-s + (1.85 + 1.74i)7-s + (2.64 − 1.00i)8-s + (−1.17 + 2.76i)9-s + (−1.98 − 0.712i)10-s + (−0.248 − 0.495i)11-s + (−3.30 + 1.03i)12-s + (2.59 − 1.11i)13-s + (0.402 − 3.57i)14-s + (2.37 + 1.01i)15-s + (−3.37 − 2.13i)16-s + (−2.14 − 2.55i)17-s + ⋯
L(s)  = 1  + (−0.600 − 0.799i)2-s + (0.552 + 0.833i)3-s + (−0.278 + 0.960i)4-s + (0.557 − 0.366i)5-s + (0.335 − 0.942i)6-s + (0.700 + 0.660i)7-s + (0.935 − 0.354i)8-s + (−0.390 + 0.920i)9-s + (−0.627 − 0.225i)10-s + (−0.0749 − 0.149i)11-s + (−0.954 + 0.298i)12-s + (0.719 − 0.310i)13-s + (0.107 − 0.956i)14-s + (0.612 + 0.262i)15-s + (−0.844 − 0.534i)16-s + (−0.520 − 0.620i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 - 0.347i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.937 - 0.347i$
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: \((2,\ 648,\ (\ :1/2),\ 0.937 - 0.347i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51017 + 0.270941i\)
\(L(\frac12)\) \(\approx\) \(1.51017 + 0.270941i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.849 + 1.13i)T \)
3 \( 1 + (-0.956 - 1.44i)T \)
good5 \( 1 + (-1.24 + 0.819i)T + (1.98 - 4.59i)T^{2} \)
7 \( 1 + (-1.85 - 1.74i)T + (0.407 + 6.98i)T^{2} \)
11 \( 1 + (0.248 + 0.495i)T + (-6.56 + 8.82i)T^{2} \)
13 \( 1 + (-2.59 + 1.11i)T + (8.92 - 9.45i)T^{2} \)
17 \( 1 + (2.14 + 2.55i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (-6.16 - 5.17i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (1.36 + 1.44i)T + (-1.33 + 22.9i)T^{2} \)
29 \( 1 + (1.03 - 0.121i)T + (28.2 - 6.68i)T^{2} \)
31 \( 1 + (1.01 + 4.26i)T + (-27.7 + 13.9i)T^{2} \)
37 \( 1 + (5.17 - 0.912i)T + (34.7 - 12.6i)T^{2} \)
41 \( 1 + (-7.75 - 5.77i)T + (11.7 + 39.2i)T^{2} \)
43 \( 1 + (0.429 - 7.37i)T + (-42.7 - 4.99i)T^{2} \)
47 \( 1 + (-12.2 - 2.91i)T + (42.0 + 21.0i)T^{2} \)
53 \( 1 + (2.74 + 4.75i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.23 + 4.45i)T + (-35.2 - 47.3i)T^{2} \)
61 \( 1 + (8.49 - 2.54i)T + (50.9 - 33.5i)T^{2} \)
67 \( 1 + (1.92 + 0.225i)T + (65.1 + 15.4i)T^{2} \)
71 \( 1 + (0.0766 + 0.0278i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-6.28 + 2.28i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-8.48 + 6.31i)T + (22.6 - 75.6i)T^{2} \)
83 \( 1 + (2.15 - 1.60i)T + (23.8 - 79.5i)T^{2} \)
89 \( 1 + (0.953 + 2.61i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-7.37 - 4.85i)T + (38.4 + 89.0i)T^{2} \)
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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

−10.53395503895409936483161717852, −9.525866691535740448161108182800, −9.202530475766890967063824458916, −8.229375397348527566788180669918, −7.66864377634042463632410066995, −5.80048563381999592402353292300, −4.93194423398761293805133324445, −3.79699787901387132846350058685, −2.72831225988230005331514973751, −1.58026657224470588844030682903, 1.10740649760175837945009533836, 2.24704712187265023316798876611, 3.96790255135908023855350882496, 5.36000886753011578710486312848, 6.32216654882194140218370296061, 7.16441968463273088941959058717, 7.68284467242787149725434985099, 8.761163842246254059419089534691, 9.266915945974797572934462627117, 10.45253968787010615770655225833

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