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Properties

Label 2-12e3-36.11-c1-0-13
Degree $2$
Conductor $1728$
Sign $0.342 + 0.939i$
Analytic cond. $13.7981$
Root an. cond. $3.71458$
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  + (−3 + 1.73i)5-s + (−3 − 1.73i)7-s + (−1.5 + 2.59i)11-s + (2 + 3.46i)13-s + 1.73i·17-s − 1.73i·19-s + (3.5 − 6.06i)25-s + (−3 − 1.73i)29-s + 12·35-s − 2·37-s + (4.5 − 2.59i)41-s + (−4.5 − 2.59i)43-s + (6 − 10.3i)47-s + (2.5 + 4.33i)49-s − 10.3i·55-s + ⋯
L(s)  = 1  + (−1.34 + 0.774i)5-s + (−1.13 − 0.654i)7-s + (−0.452 + 0.783i)11-s + (0.554 + 0.960i)13-s + 0.420i·17-s − 0.397i·19-s + (0.700 − 1.21i)25-s + (−0.557 − 0.321i)29-s + 2.02·35-s − 0.328·37-s + (0.702 − 0.405i)41-s + (−0.686 − 0.396i)43-s + (0.875 − 1.51i)47-s + (0.357 + 0.618i)49-s − 1.40i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.342 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5243389488\)
\(L(\frac12)\) \(\approx\) \(0.5243389488\)
\(L(\frac{3}{2})\) 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 \( 1 + (3 - 1.73i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (3 + 1.73i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 1.73iT - 17T^{2} \)
19 \( 1 + 1.73iT - 19T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 1.73i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 2T + 37T^{2} \)
41 \( 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.5 + 2.59i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (7.5 + 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 11T + 73T^{2} \)
79 \( 1 + (-3 - 1.73i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 13.8iT - 89T^{2} \)
97 \( 1 + (6.5 - 11.2i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.225372151434558477169559033305, −8.247318681222401571380142768445, −7.39782548922027803675663333441, −6.91492001838096773895601695698, −6.26812505487645994004940357050, −4.85975193573797985228309119711, −3.80791135278475011820719256269, −3.53485522255024667314453387194, −2.19607955823744858583892575526, −0.26514203081634512701661242326, 0.888138492619801174951863627475, 2.86203641804073822991278420309, 3.45949075422610355847758834718, 4.42746028408117879293741535824, 5.53039100992771383728790366875, 6.07330681349118135528612935377, 7.28827038735117538891449799997, 7.989117999096146846008441345305, 8.655598750842081782487083748958, 9.253542531866835918691305774595

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