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Properties

Label 2-648-216.13-c1-0-22
Degree $2$
Conductor $648$
Sign $0.0102 + 0.999i$
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  + (−0.289 − 1.38i)2-s + (−1.83 + 0.801i)4-s + (1.55 − 1.85i)5-s + (0.763 + 0.278i)7-s + (1.63 + 2.30i)8-s + (−3.02 − 1.61i)10-s + (2.23 + 2.65i)11-s + (2.83 + 0.500i)13-s + (0.163 − 1.13i)14-s + (2.71 − 2.93i)16-s + (3.55 − 6.15i)17-s + (−5.16 + 2.98i)19-s + (−1.36 + 4.65i)20-s + (3.03 − 3.85i)22-s + (6.58 − 2.39i)23-s + ⋯
L(s)  = 1  + (−0.204 − 0.978i)2-s + (−0.916 + 0.400i)4-s + (0.696 − 0.830i)5-s + (0.288 + 0.105i)7-s + (0.579 + 0.814i)8-s + (−0.955 − 0.511i)10-s + (0.672 + 0.801i)11-s + (0.787 + 0.138i)13-s + (0.0437 − 0.304i)14-s + (0.678 − 0.734i)16-s + (0.861 − 1.49i)17-s + (−1.18 + 0.683i)19-s + (−0.305 + 1.03i)20-s + (0.646 − 0.822i)22-s + (1.37 − 0.500i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0102 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0102 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.0102 + 0.999i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.0102 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08427 - 1.07316i\)
\(L(\frac12)\) \(\approx\) \(1.08427 - 1.07316i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.289 + 1.38i)T \)
3 \( 1 \)
good5 \( 1 + (-1.55 + 1.85i)T + (-0.868 - 4.92i)T^{2} \)
7 \( 1 + (-0.763 - 0.278i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (-2.23 - 2.65i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (-2.83 - 0.500i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (-3.55 + 6.15i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.16 - 2.98i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.58 + 2.39i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (6.59 - 1.16i)T + (27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2.55 + 0.930i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-3.59 - 2.07i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.742 + 4.20i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (1.77 + 2.10i)T + (-7.46 + 42.3i)T^{2} \)
47 \( 1 + (2.80 + 1.01i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 7.38iT - 53T^{2} \)
59 \( 1 + (-4.72 + 5.62i)T + (-10.2 - 58.1i)T^{2} \)
61 \( 1 + (-0.0900 + 0.247i)T + (-46.7 - 39.2i)T^{2} \)
67 \( 1 + (-2.20 - 0.388i)T + (62.9 + 22.9i)T^{2} \)
71 \( 1 + (2.69 - 4.67i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.97 + 3.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.76 - 10.0i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (7.86 - 1.38i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (2.14 + 3.72i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.28 + 3.59i)T + (16.8 - 95.5i)T^{2} \)
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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

−10.24833838601041835669484446869, −9.454970767709875234666712599456, −8.947485185828662199868104642211, −8.058731404676124689341177524850, −6.82864003767133472192921407532, −5.43779515892634525870557456005, −4.73366514133917495369862504291, −3.61747775463205296341598132166, −2.15111500184829101408101711384, −1.12883286150674918639953041280, 1.39458883857835081544388759895, 3.26999046925849254542774209148, 4.38198754812947627407077164362, 5.84408129226111095675584430719, 6.17049655312142469876915846993, 7.11229365976858039642466859282, 8.188790101656126585306599457382, 8.864024372298489961749770825094, 9.789368226454290484555875909212, 10.73571903543584603275588109476

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