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Properties

Label 2-432-36.7-c2-0-9
Degree $2$
Conductor $432$
Sign $-0.416 + 0.909i$
Analytic cond. $11.7711$
Root an. cond. $3.43091$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.355 − 0.615i)5-s + (2.70 + 1.56i)7-s + (−14.3 − 8.30i)11-s + (−9.17 − 15.8i)13-s + 9.69·17-s + 8.20i·19-s + (−1.94 + 1.12i)23-s + (12.2 − 21.2i)25-s + (20.8 − 36.0i)29-s + (−21.6 + 12.4i)31-s − 2.21i·35-s − 40.3·37-s + (−25.6 − 44.5i)41-s + (−56.6 − 32.7i)43-s + (−29.2 − 16.9i)47-s + ⋯
L(s)  = 1  + (−0.0710 − 0.123i)5-s + (0.386 + 0.223i)7-s + (−1.30 − 0.754i)11-s + (−0.705 − 1.22i)13-s + 0.570·17-s + 0.431i·19-s + (−0.0847 + 0.0489i)23-s + (0.489 − 0.848i)25-s + (0.717 − 1.24i)29-s + (−0.697 + 0.402i)31-s − 0.0634i·35-s − 1.09·37-s + (−0.626 − 1.08i)41-s + (−1.31 − 0.760i)43-s + (−0.623 − 0.359i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.416 + 0.909i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.416 + 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.416 + 0.909i$
Analytic conductor: \(11.7711\)
Root analytic conductor: \(3.43091\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1),\ -0.416 + 0.909i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.545173 - 0.849419i\)
\(L(\frac12)\) \(\approx\) \(0.545173 - 0.849419i\)
\(L(2)\) 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 \( 1 + (0.355 + 0.615i)T + (-12.5 + 21.6i)T^{2} \)
7 \( 1 + (-2.70 - 1.56i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (14.3 + 8.30i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (9.17 + 15.8i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 - 9.69T + 289T^{2} \)
19 \( 1 - 8.20iT - 361T^{2} \)
23 \( 1 + (1.94 - 1.12i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-20.8 + 36.0i)T + (-420.5 - 728. i)T^{2} \)
31 \( 1 + (21.6 - 12.4i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 + 40.3T + 1.36e3T^{2} \)
41 \( 1 + (25.6 + 44.5i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (56.6 + 32.7i)T + (924.5 + 1.60e3i)T^{2} \)
47 \( 1 + (29.2 + 16.9i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 90.6T + 2.80e3T^{2} \)
59 \( 1 + (-66.2 + 38.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-1.35 + 2.35i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (34.5 - 19.9i)T + (2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 102. iT - 5.04e3T^{2} \)
73 \( 1 - 38.1T + 5.32e3T^{2} \)
79 \( 1 + (-94.4 - 54.5i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (113. + 65.5i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 38.0T + 7.92e3T^{2} \)
97 \( 1 + (12.1 - 21.1i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−10.33959040461848636918952243577, −10.16832257540151585442194385504, −8.465697933027110240823138901064, −8.155695753558831766539264454593, −7.03244973784019366299371622917, −5.57004377541655348240109517117, −5.12590359216740833388925312914, −3.50273486350006134937827394832, −2.36079937650791214244049160451, −0.40999232143660058919783808648, 1.75188418638315916826231812505, 3.08345385762776345279798165628, 4.60997560258312362199287613025, 5.24622829029199026706081438189, 6.81780468082725976807494880802, 7.42519673645493383508983406013, 8.449439887806872338880933787142, 9.534421615013693308389006797798, 10.32032371751277795655079275949, 11.19382339283798084280544322895

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