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Properties

Label 2-648-216.59-c1-0-13
Degree $2$
Conductor $648$
Sign $0.363 + 0.931i$
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  + (−0.860 − 1.12i)2-s + (−0.517 + 1.93i)4-s + (−3.00 + 1.09i)5-s + (−4.58 + 0.808i)7-s + (2.61 − 1.08i)8-s + (3.81 + 2.43i)10-s + (0.303 − 0.833i)11-s + (1.22 − 1.45i)13-s + (4.85 + 4.44i)14-s + (−3.46 − 2.00i)16-s + (4.03 + 2.32i)17-s + (−0.171 − 0.296i)19-s + (−0.557 − 6.38i)20-s + (−1.19 + 0.377i)22-s + (1.00 − 5.68i)23-s + ⋯
L(s)  = 1  + (−0.608 − 0.793i)2-s + (−0.258 + 0.965i)4-s + (−1.34 + 0.489i)5-s + (−1.73 + 0.305i)7-s + (0.923 − 0.382i)8-s + (1.20 + 0.769i)10-s + (0.0914 − 0.251i)11-s + (0.338 − 0.403i)13-s + (1.29 + 1.18i)14-s + (−0.865 − 0.500i)16-s + (0.978 + 0.564i)17-s + (−0.0393 − 0.0680i)19-s + (−0.124 − 1.42i)20-s + (−0.255 + 0.0804i)22-s + (0.208 − 1.18i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.363 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.363 + 0.931i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.363 + 0.931i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.446184 - 0.304842i\)
\(L(\frac12)\) \(\approx\) \(0.446184 - 0.304842i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.860 + 1.12i)T \)
3 \( 1 \)
good5 \( 1 + (3.00 - 1.09i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (4.58 - 0.808i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-0.303 + 0.833i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-1.22 + 1.45i)T + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-4.03 - 2.32i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.171 + 0.296i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.00 + 5.68i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-4.16 + 3.49i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-7.80 - 1.37i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (2.31 + 1.33i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.0346 - 0.0413i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (2.85 + 1.04i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.90 - 10.7i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 1.27T + 53T^{2} \)
59 \( 1 + (2.43 + 6.68i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (-8.19 + 1.44i)T + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (6.31 + 5.29i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-0.186 + 0.323i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.29 - 10.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.54 + 3.03i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (8.90 + 10.6i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (-6.35 + 3.66i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.833 + 0.303i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.34809197262659168814391098887, −9.750282117989154717824704393876, −8.614686666253704821858873936092, −8.027011264691121835101426309391, −6.98010414013642287910627366201, −6.16340854054302754338677177126, −4.33886954408128290523170204525, −3.35811187280104389067237375732, −2.87319030342541885804491604968, −0.54060594109632515994553817060, 0.854718948687265918084925336217, 3.26404238537842774642155024614, 4.22805941282941235709750108785, 5.41074238948402045472569749279, 6.58728673655879807495114724045, 7.18194623539826934935811887022, 8.048888794294833802945390362793, 8.923002973510754177590599792987, 9.726376591424493426257816765024, 10.37075684897174748696978813143

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