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Properties

Label 2-12e3-9.7-c1-0-8
Degree $2$
Conductor $1728$
Sign $0.939 - 0.342i$
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  + (1 − 1.73i)7-s + (−1.5 + 2.59i)11-s + (1 + 1.73i)13-s + 3·17-s − 19-s + (3 + 5.19i)23-s + (2.5 − 4.33i)25-s + (−3 + 5.19i)29-s + (−2 − 3.46i)31-s + 4·37-s + (4.5 + 7.79i)41-s + (0.5 − 0.866i)43-s + (3 − 5.19i)47-s + (1.50 + 2.59i)49-s + 12·53-s + ⋯
L(s)  = 1  + (0.377 − 0.654i)7-s + (−0.452 + 0.783i)11-s + (0.277 + 0.480i)13-s + 0.727·17-s − 0.229·19-s + (0.625 + 1.08i)23-s + (0.5 − 0.866i)25-s + (−0.557 + 0.964i)29-s + (−0.359 − 0.622i)31-s + 0.657·37-s + (0.702 + 1.21i)41-s + (0.0762 − 0.132i)43-s + (0.437 − 0.757i)47-s + (0.214 + 0.371i)49-s + 1.64·53-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (1153, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.939 - 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.786118994\)
\(L(\frac12)\) \(\approx\) \(1.786118994\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1 + 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 3T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (-4.5 - 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (-1.5 - 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-6 + 10.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
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

−9.413952841324163097615930753077, −8.575093211109862766728822233167, −7.57755437851135847301686674868, −7.23350114385932295842075927388, −6.15541313084689976978420961095, −5.19762583583069829640978499542, −4.40765636839452239207921215767, −3.51825212912594374608338265663, −2.27513085812152785197533481898, −1.09102071250634604603810883828, 0.841618103715855155454521688878, 2.33240302370886618173630476921, 3.19951819318163394143757783228, 4.28871165727223884298952398795, 5.48933806763423039282282233877, 5.73707815255641406110948342675, 6.94946662572253571492045894966, 7.79042020656272212454085421325, 8.571954221164352986571117239136, 9.057524599899958380258212951267

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