Properties

Label 2-648-648.155-c1-0-39
Degree $2$
Conductor $648$
Sign $0.527 - 0.849i$
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  + (−0.392 + 1.35i)2-s + (−1.36 + 1.05i)3-s + (−1.69 − 1.06i)4-s + (−0.991 + 0.652i)5-s + (−0.902 − 2.27i)6-s + (−0.145 − 0.137i)7-s + (2.11 − 1.88i)8-s + (0.753 − 2.90i)9-s + (−0.497 − 1.60i)10-s + (−0.414 − 0.824i)11-s + (3.44 − 0.332i)12-s + (5.37 − 2.31i)13-s + (0.243 − 0.143i)14-s + (0.667 − 1.94i)15-s + (1.72 + 3.60i)16-s + (−3.81 − 4.55i)17-s + ⋯
L(s)  = 1  + (−0.277 + 0.960i)2-s + (−0.790 + 0.611i)3-s + (−0.845 − 0.533i)4-s + (−0.443 + 0.291i)5-s + (−0.368 − 0.929i)6-s + (−0.0549 − 0.0518i)7-s + (0.747 − 0.664i)8-s + (0.251 − 0.967i)9-s + (−0.157 − 0.507i)10-s + (−0.124 − 0.248i)11-s + (0.995 − 0.0959i)12-s + (1.48 − 0.642i)13-s + (0.0650 − 0.0384i)14-s + (0.172 − 0.502i)15-s + (0.431 + 0.902i)16-s + (−0.925 − 1.10i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.527 - 0.849i$
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: \((2,\ 648,\ (\ :1/2),\ 0.527 - 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.694170 + 0.385860i\)
\(L(\frac12)\) \(\approx\) \(0.694170 + 0.385860i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.392 - 1.35i)T \)
3 \( 1 + (1.36 - 1.05i)T \)
good5 \( 1 + (0.991 - 0.652i)T + (1.98 - 4.59i)T^{2} \)
7 \( 1 + (0.145 + 0.137i)T + (0.407 + 6.98i)T^{2} \)
11 \( 1 + (0.414 + 0.824i)T + (-6.56 + 8.82i)T^{2} \)
13 \( 1 + (-5.37 + 2.31i)T + (8.92 - 9.45i)T^{2} \)
17 \( 1 + (3.81 + 4.55i)T + (-2.95 + 16.7i)T^{2} \)
19 \( 1 + (-3.38 - 2.84i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (2.12 + 2.24i)T + (-1.33 + 22.9i)T^{2} \)
29 \( 1 + (-4.36 + 0.509i)T + (28.2 - 6.68i)T^{2} \)
31 \( 1 + (-0.0796 - 0.335i)T + (-27.7 + 13.9i)T^{2} \)
37 \( 1 + (-6.35 + 1.11i)T + (34.7 - 12.6i)T^{2} \)
41 \( 1 + (3.38 + 2.51i)T + (11.7 + 39.2i)T^{2} \)
43 \( 1 + (0.194 - 3.33i)T + (-42.7 - 4.99i)T^{2} \)
47 \( 1 + (-9.46 - 2.24i)T + (42.0 + 21.0i)T^{2} \)
53 \( 1 + (-3.23 - 5.61i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.664 - 1.32i)T + (-35.2 - 47.3i)T^{2} \)
61 \( 1 + (-4.70 + 1.40i)T + (50.9 - 33.5i)T^{2} \)
67 \( 1 + (7.00 + 0.818i)T + (65.1 + 15.4i)T^{2} \)
71 \( 1 + (-5.08 - 1.84i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (-15.3 + 5.59i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-11.0 + 8.22i)T + (22.6 - 75.6i)T^{2} \)
83 \( 1 + (5.74 - 4.28i)T + (23.8 - 79.5i)T^{2} \)
89 \( 1 + (1.72 + 4.72i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (6.92 + 4.55i)T + (38.4 + 89.0i)T^{2} \)
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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

−10.67351249243729098330540078421, −9.778823845888256854043680408947, −8.932108468732292972526896503872, −8.033764351969960211998505390911, −7.04214403392885459620990051816, −6.16396302915593666310151072180, −5.45743831220991481204235182442, −4.37794660019542297000495969034, −3.44204504864019861410328070300, −0.74353376771293987660133151993, 0.985788688911148778112385834367, 2.20406239586617704078613851597, 3.83727777629463329088319421941, 4.64781909995674853634213838360, 5.88424549284822265018521387779, 6.88239613056331865087600454917, 8.057700786127639826137749217865, 8.609374803637562773406462118429, 9.710888990643707298737747393359, 10.75139147614799450745906231977

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