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Properties

Label 2-648-9.4-c1-0-7
Degree $2$
Conductor $648$
Sign $0.939 + 0.342i$
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  + (2 − 3.46i)5-s + (1.5 + 2.59i)7-s + (2 + 3.46i)11-s + (−0.5 + 0.866i)13-s + 4·17-s − 19-s + (2 − 3.46i)23-s + (−5.49 − 9.52i)25-s + (2 − 3.46i)31-s + 12·35-s − 9·37-s + (4 + 6.92i)43-s + (−6 − 10.3i)47-s + (−1 + 1.73i)49-s + 8·53-s + ⋯
L(s)  = 1  + (0.894 − 1.54i)5-s + (0.566 + 0.981i)7-s + (0.603 + 1.04i)11-s + (−0.138 + 0.240i)13-s + 0.970·17-s − 0.229·19-s + (0.417 − 0.722i)23-s + (−1.09 − 1.90i)25-s + (0.359 − 0.622i)31-s + 2.02·35-s − 1.47·37-s + (0.609 + 1.05i)43-s + (−0.875 − 1.51i)47-s + (−0.142 + 0.247i)49-s + 1.09·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(648\)    =    \(2^{3} \cdot 3^{4}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(5.17430\)
Root analytic conductor: \(2.27471\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{648} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 648,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88004 - 0.331503i\)
\(L(\frac12)\) \(\approx\) \(1.88004 - 0.331503i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-2 + 3.46i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 9T + 37T^{2} \)
41 \( 1 + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (6 + 10.3i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8T + 53T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (-2.5 - 4.33i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4 - 6.92i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12T + 89T^{2} \)
97 \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.20431382276655465882338804936, −9.546007168045528392149126199841, −8.798398343184649507679526160247, −8.227797685384042246310201623796, −6.88759238811060823009727248613, −5.70605435630144858491743659004, −5.10465732902371847600522083362, −4.24661457154967422821025588463, −2.30909798992084527564995644954, −1.37128675713808375905464245188, 1.43782546480098954740283640012, 2.96178613790866144012933753841, 3.75165347789939120812946174462, 5.31416337405795539530117539142, 6.20813409595174553731371670845, 7.04074678340430496644103322130, 7.75534287426012070621541069968, 8.970499562106101141178664817742, 10.01453594948184964011004898348, 10.62023007658225791457938254873

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