Properties

Label 2-648-648.155-c1-0-4
Degree $2$
Conductor $648$
Sign $0.332 + 0.942i$
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  + (0.243 + 1.39i)2-s + (−0.418 + 1.68i)3-s + (−1.88 + 0.678i)4-s + (−0.258 + 0.170i)5-s + (−2.44 − 0.172i)6-s + (0.0104 + 0.00981i)7-s + (−1.40 − 2.45i)8-s + (−2.65 − 1.40i)9-s + (−0.299 − 0.318i)10-s + (−1.41 − 2.81i)11-s + (−0.354 − 3.44i)12-s + (−2.26 + 0.978i)13-s + (−0.0111 + 0.0168i)14-s + (−0.177 − 0.505i)15-s + (3.07 − 2.55i)16-s + (−1.51 − 1.79i)17-s + ⋯
L(s)  = 1  + (0.172 + 0.985i)2-s + (−0.241 + 0.970i)3-s + (−0.940 + 0.339i)4-s + (−0.115 + 0.0760i)5-s + (−0.997 − 0.0705i)6-s + (0.00393 + 0.00371i)7-s + (−0.496 − 0.868i)8-s + (−0.883 − 0.468i)9-s + (−0.0948 − 0.100i)10-s + (−0.426 − 0.849i)11-s + (−0.102 − 0.994i)12-s + (−0.629 + 0.271i)13-s + (−0.00297 + 0.00451i)14-s + (−0.0458 − 0.130i)15-s + (0.769 − 0.638i)16-s + (−0.366 − 0.436i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.332 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.332 + 0.942i$
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: \((2,\ 648,\ (\ :1/2),\ 0.332 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0432096 - 0.0305716i\)
\(L(\frac12)\) \(\approx\) \(0.0432096 - 0.0305716i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.243 - 1.39i)T \)
3 \( 1 + (0.418 - 1.68i)T \)
good5 \( 1 + (0.258 - 0.170i)T + (1.98 - 4.59i)T^{2} \)
7 \( 1 + (-0.0104 - 0.00981i)T + (0.407 + 6.98i)T^{2} \)
11 \( 1 + (1.41 + 2.81i)T + (-6.56 + 8.82i)T^{2} \)
13 \( 1 + (2.26 - 0.978i)T + (8.92 - 9.45i)T^{2} \)
17 \( 1 + (1.51 + 1.79i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (4.81 + 4.04i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (-4.28 - 4.54i)T + (-1.33 + 22.9i)T^{2} \)
29 \( 1 + (4.55 - 0.532i)T + (28.2 - 6.68i)T^{2} \)
31 \( 1 + (0.306 + 1.29i)T + (-27.7 + 13.9i)T^{2} \)
37 \( 1 + (8.30 - 1.46i)T + (34.7 - 12.6i)T^{2} \)
41 \( 1 + (-1.30 - 0.973i)T + (11.7 + 39.2i)T^{2} \)
43 \( 1 + (-0.0545 + 0.936i)T + (-42.7 - 4.99i)T^{2} \)
47 \( 1 + (-3.27 - 0.777i)T + (42.0 + 21.0i)T^{2} \)
53 \( 1 + (-6.75 - 11.6i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.50 + 6.98i)T + (-35.2 - 47.3i)T^{2} \)
61 \( 1 + (8.08 - 2.42i)T + (50.9 - 33.5i)T^{2} \)
67 \( 1 + (5.45 + 0.637i)T + (65.1 + 15.4i)T^{2} \)
71 \( 1 + (6.56 + 2.39i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (12.6 - 4.59i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-7.89 + 5.87i)T + (22.6 - 75.6i)T^{2} \)
83 \( 1 + (1.94 - 1.44i)T + (23.8 - 79.5i)T^{2} \)
89 \( 1 + (0.233 + 0.640i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (-5.47 - 3.60i)T + (38.4 + 89.0i)T^{2} \)
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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

−11.14438006896383296659589215224, −10.31970417178998240861111296685, −9.112831742161136238643298915460, −8.940520187672153873539722264311, −7.65007514113301248124617679946, −6.78369102937831444377144138861, −5.67305659932684259380357591231, −5.03210289122636835119051090880, −4.04044126975008925005545735654, −2.99631919692961022720629687056, 0.02668885467223287058548114453, 1.78297208934857903172488547905, 2.63860077082993169132506710056, 4.14459204287433009102224227327, 5.14270825804658020369664597709, 6.14480177685597108349129210525, 7.25839079425770983355174691708, 8.223730377583416073545979743278, 8.957988991904772814246761194552, 10.26730883259682795782383437606

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