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Properties

Label 2-12e3-9.4-c1-0-7
Degree $2$
Conductor $1728$
Sign $0.766 - 0.642i$
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  + (0.5 − 0.866i)5-s + (1.5 + 2.59i)7-s + (−2.5 − 4.33i)11-s + (−2.5 + 4.33i)13-s + 2·17-s + 4·19-s + (−0.5 + 0.866i)23-s + (2 + 3.46i)25-s + (4.5 + 7.79i)29-s + (0.5 − 0.866i)31-s + 3·35-s + 6·37-s + (1.5 − 2.59i)41-s + (0.5 + 0.866i)43-s + (−1.5 − 2.59i)47-s + ⋯
L(s)  = 1  + (0.223 − 0.387i)5-s + (0.566 + 0.981i)7-s + (−0.753 − 1.30i)11-s + (−0.693 + 1.20i)13-s + 0.485·17-s + 0.917·19-s + (−0.104 + 0.180i)23-s + (0.400 + 0.692i)25-s + (0.835 + 1.44i)29-s + (0.0898 − 0.155i)31-s + 0.507·35-s + 0.986·37-s + (0.234 − 0.405i)41-s + (0.0762 + 0.132i)43-s + (−0.218 − 0.378i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1728\)    =    \(2^{6} \cdot 3^{3}\)
Sign: $0.766 - 0.642i$
Analytic conductor: \(13.7981\)
Root analytic conductor: \(3.71458\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1728} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1728,\ (\ :1/2),\ 0.766 - 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.777963294\)
\(L(\frac12)\) \(\approx\) \(1.777963294\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.5 - 4.33i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 - 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + (5.5 - 9.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 18T + 89T^{2} \)
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

−9.125793295497294670814919804552, −8.814670618367569646028829830722, −7.914012287137553660015379919875, −7.11280994275382709040086144947, −5.95753065724749892998102676273, −5.35236108842587891873169991266, −4.69724642288974768102775052680, −3.31861385924957063349058800128, −2.42785416843290223160096242374, −1.19909660639261215221639309868, 0.76882466295312432264096735695, 2.24771920356935473319912296767, 3.11963436013871109334904510300, 4.45196926425254274734800207554, 4.95739148008411474292527925225, 6.00948852343102586575430579563, 7.02962111817897195154316627126, 7.79915518866265837920495041644, 7.991496866410054745167653272593, 9.543731381351568004749203531654

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