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Properties

Label 2-648-216.11-c1-0-18
Degree $2$
Conductor $648$
Sign $0.988 - 0.150i$
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  + (1.41 − 0.103i)2-s + (1.97 − 0.291i)4-s + (2.25 + 0.819i)5-s + (−4.05 − 0.715i)7-s + (2.76 − 0.615i)8-s + (3.25 + 0.923i)10-s + (1.59 + 4.39i)11-s + (2.03 + 2.42i)13-s + (−5.79 − 0.589i)14-s + (3.83 − 1.15i)16-s + (5.30 − 3.06i)17-s + (1.49 − 2.58i)19-s + (4.69 + 0.965i)20-s + (2.70 + 6.03i)22-s + (−0.264 − 1.49i)23-s + ⋯
L(s)  = 1  + (0.997 − 0.0730i)2-s + (0.989 − 0.145i)4-s + (1.00 + 0.366i)5-s + (−1.53 − 0.270i)7-s + (0.976 − 0.217i)8-s + (1.03 + 0.291i)10-s + (0.482 + 1.32i)11-s + (0.564 + 0.672i)13-s + (−1.54 − 0.157i)14-s + (0.957 − 0.288i)16-s + (1.28 − 0.743i)17-s + (0.342 − 0.593i)19-s + (1.04 + 0.215i)20-s + (0.577 + 1.28i)22-s + (−0.0550 − 0.312i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.988 - 0.150i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.988 - 0.150i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.01136 + 0.228113i\)
\(L(\frac12)\) \(\approx\) \(3.01136 + 0.228113i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.41 + 0.103i)T \)
3 \( 1 \)
good5 \( 1 + (-2.25 - 0.819i)T + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (4.05 + 0.715i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-1.59 - 4.39i)T + (-8.42 + 7.07i)T^{2} \)
13 \( 1 + (-2.03 - 2.42i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (-5.30 + 3.06i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.49 + 2.58i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.264 + 1.49i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (1.88 + 1.58i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (8.06 - 1.42i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-0.515 + 0.297i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.89 + 5.82i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (4.63 - 1.68i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.750 - 4.25i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 6.46T + 53T^{2} \)
59 \( 1 + (-0.703 + 1.93i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (7.44 + 1.31i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-5.64 + 4.73i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (1.78 + 3.08i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.85 - 10.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.66 - 7.94i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-5.79 + 6.90i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-7.85 - 4.53i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.05 - 1.47i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.49494393012946278467272261383, −9.755663625672526694573935818021, −9.340792230119814805895145130510, −7.41286194346902372919519600983, −6.77670440396128990967264133502, −6.13604465413740161654499360439, −5.14192057100647538949158448294, −3.88731488947944141785710075505, −2.97528943047996547955231242715, −1.76642440302625141238580248229, 1.51378973807290844355945387130, 3.24902119552035183152736891268, 3.55433816768743993725143181537, 5.49379668661086025827701937256, 5.83479779485122077147293991633, 6.47842938481464963657216921772, 7.78695811813015910677301949868, 8.879232840834160164853884801200, 9.827307289980004445256815602730, 10.48909974167914895975793258470

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