Properties

Label 2-648-648.133-c1-0-17
Degree $2$
Conductor $648$
Sign $-0.672 + 0.740i$
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  + (0.876 + 1.11i)2-s + (−0.817 + 1.52i)3-s + (−0.465 + 1.94i)4-s + (0.00150 − 0.00141i)5-s + (−2.41 + 0.430i)6-s + (−0.227 + 0.0266i)7-s + (−2.56 + 1.18i)8-s + (−1.66 − 2.49i)9-s + (0.00289 + 0.000426i)10-s + (−1.22 + 0.368i)11-s + (−2.59 − 2.30i)12-s + (−4.36 + 0.254i)13-s + (−0.229 − 0.229i)14-s + (0.000937 + 0.00345i)15-s + (−3.56 − 1.80i)16-s + (−3.40 − 1.24i)17-s + ⋯
L(s)  = 1  + (0.619 + 0.785i)2-s + (−0.471 + 0.881i)3-s + (−0.232 + 0.972i)4-s + (0.000672 − 0.000634i)5-s + (−0.984 + 0.175i)6-s + (−0.0860 + 0.0100i)7-s + (−0.907 + 0.419i)8-s + (−0.554 − 0.832i)9-s + (0.000915 + 0.000134i)10-s + (−0.370 + 0.111i)11-s + (−0.747 − 0.664i)12-s + (−1.20 + 0.0704i)13-s + (−0.0612 − 0.0613i)14-s + (0.000242 + 0.000892i)15-s + (−0.891 − 0.452i)16-s + (−0.826 − 0.300i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.672 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $-0.672 + 0.740i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ -0.672 + 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.339346 - 0.766920i\)
\(L(\frac12)\) \(\approx\) \(0.339346 - 0.766920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.876 - 1.11i)T \)
3 \( 1 + (0.817 - 1.52i)T \)
good5 \( 1 + (-0.00150 + 0.00141i)T + (0.290 - 4.99i)T^{2} \)
7 \( 1 + (0.227 - 0.0266i)T + (6.81 - 1.61i)T^{2} \)
11 \( 1 + (1.22 - 0.368i)T + (9.19 - 6.04i)T^{2} \)
13 \( 1 + (4.36 - 0.254i)T + (12.9 - 1.50i)T^{2} \)
17 \( 1 + (3.40 + 1.24i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (-0.999 - 2.74i)T + (-14.5 + 12.2i)T^{2} \)
23 \( 1 + (-3.60 - 0.421i)T + (22.3 + 5.30i)T^{2} \)
29 \( 1 + (-1.10 - 2.20i)T + (-17.3 + 23.2i)T^{2} \)
31 \( 1 + (0.0468 - 0.0629i)T + (-8.89 - 29.6i)T^{2} \)
37 \( 1 + (0.538 + 0.641i)T + (-6.42 + 36.4i)T^{2} \)
41 \( 1 + (5.74 - 3.77i)T + (16.2 - 37.6i)T^{2} \)
43 \( 1 + (-0.284 + 1.20i)T + (-38.4 - 19.2i)T^{2} \)
47 \( 1 + (0.816 + 1.09i)T + (-13.4 + 45.0i)T^{2} \)
53 \( 1 + (-7.12 + 4.11i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.96 - 2.08i)T + (49.2 + 32.4i)T^{2} \)
61 \( 1 + (0.693 + 0.299i)T + (41.8 + 44.3i)T^{2} \)
67 \( 1 + (5.83 - 11.6i)T + (-40.0 - 53.7i)T^{2} \)
71 \( 1 + (-0.402 - 2.28i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (1.30 - 7.41i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-9.43 - 6.20i)T + (31.2 + 72.5i)T^{2} \)
83 \( 1 + (-0.822 + 1.25i)T + (-32.8 - 76.2i)T^{2} \)
89 \( 1 + (-0.854 + 4.84i)T + (-83.6 - 30.4i)T^{2} \)
97 \( 1 + (5.49 - 5.82i)T + (-5.64 - 96.8i)T^{2} \)
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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

−11.25575603051998823178670132617, −10.13639900961713214101581152368, −9.353705177857470326529965860858, −8.502022620729246999261108469780, −7.33609297179727038655236937619, −6.60495417026946271434966876797, −5.38997595113617834315065627701, −4.94109526594303235107952549083, −3.86722738240065589774642789950, −2.76959165628905205620307489948, 0.36779483996049428331819634944, 2.03878502833286322315649917408, 2.92539788111550685725978776423, 4.55833560868218154343104726925, 5.26324276916823801910664447187, 6.35191828767824969668307750584, 7.08427027311933964247263914004, 8.255302640542933555414451594544, 9.317391843128243927056832591436, 10.35371779269158304955881941535

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