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Properties

Label 2-432-36.23-c1-0-2
Degree $2$
Conductor $432$
Sign $0.984 - 0.173i$
Analytic cond. $3.44953$
Root an. cond. $1.85729$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82414 + 0.159592i\)
\(L(\frac12)\) \(\approx\) \(1.82414 + 0.159592i\)
\(L(\frac{3}{2})\) 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

−11.07176264594324387013343194173, −10.29988898878329244917503481632, −9.578784711289756383359066857849, −8.431880234787505341214263221764, −7.40690618741650001866817120147, −6.48228486428504359883928337938, −5.49725288943448370758910298446, −4.44532784685833394022858043204, −2.82790684700660830506267903358, −1.65220790002750827038030857601, 1.57542735696016145572239816235, 2.59467094678548112024010140714, 4.75018633803456829721590010696, 5.21924573534387190387217921712, 6.17499877304241357026965110457, 7.62672481306386818099620522974, 8.458084611072879884755543794328, 9.359795178654280035191976249498, 10.12470483992920131287663712728, 11.01963095866036898035130123639

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