Properties

Label 2-432-9.4-c7-0-13
Degree $2$
Conductor $432$
Sign $-0.633 - 0.774i$
Analytic cond. $134.950$
Root an. cond. $11.6168$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−47.9 + 83.1i)5-s + (189. + 327. i)7-s + (3.43e3 + 5.95e3i)11-s + (4.82e3 − 8.35e3i)13-s − 2.14e4·17-s − 5.51e3·19-s + (−3.14e4 + 5.45e4i)23-s + (3.44e4 + 5.96e4i)25-s + (1.11e5 + 1.92e5i)29-s + (5.77e4 − 9.99e4i)31-s − 3.62e4·35-s + 8.17e4·37-s + (2.98e5 − 5.17e5i)41-s + (3.38e4 + 5.86e4i)43-s + (−1.51e5 − 2.62e5i)47-s + ⋯
L(s)  = 1  + (−0.171 + 0.297i)5-s + (0.208 + 0.360i)7-s + (0.778 + 1.34i)11-s + (0.609 − 1.05i)13-s − 1.05·17-s − 0.184·19-s + (−0.539 + 0.934i)23-s + (0.441 + 0.763i)25-s + (0.846 + 1.46i)29-s + (0.348 − 0.602i)31-s − 0.143·35-s + 0.265·37-s + (0.677 − 1.17i)41-s + (0.0649 + 0.112i)43-s + (−0.213 − 0.369i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.633 - 0.774i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.633 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.633 - 0.774i$
Analytic conductor: \(134.950\)
Root analytic conductor: \(11.6168\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :7/2),\ -0.633 - 0.774i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.641613131\)
\(L(\frac12)\) \(\approx\) \(1.641613131\)
\(L(\frac{9}{2})\) 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 \( 1 + (47.9 - 83.1i)T + (-3.90e4 - 6.76e4i)T^{2} \)
7 \( 1 + (-189. - 327. i)T + (-4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (-3.43e3 - 5.95e3i)T + (-9.74e6 + 1.68e7i)T^{2} \)
13 \( 1 + (-4.82e3 + 8.35e3i)T + (-3.13e7 - 5.43e7i)T^{2} \)
17 \( 1 + 2.14e4T + 4.10e8T^{2} \)
19 \( 1 + 5.51e3T + 8.93e8T^{2} \)
23 \( 1 + (3.14e4 - 5.45e4i)T + (-1.70e9 - 2.94e9i)T^{2} \)
29 \( 1 + (-1.11e5 - 1.92e5i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + (-5.77e4 + 9.99e4i)T + (-1.37e10 - 2.38e10i)T^{2} \)
37 \( 1 - 8.17e4T + 9.49e10T^{2} \)
41 \( 1 + (-2.98e5 + 5.17e5i)T + (-9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (-3.38e4 - 5.86e4i)T + (-1.35e11 + 2.35e11i)T^{2} \)
47 \( 1 + (1.51e5 + 2.62e5i)T + (-2.53e11 + 4.38e11i)T^{2} \)
53 \( 1 + 8.46e5T + 1.17e12T^{2} \)
59 \( 1 + (7.93e5 - 1.37e6i)T + (-1.24e12 - 2.15e12i)T^{2} \)
61 \( 1 + (-1.12e6 - 1.95e6i)T + (-1.57e12 + 2.72e12i)T^{2} \)
67 \( 1 + (-1.51e6 + 2.61e6i)T + (-3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + 4.41e6T + 9.09e12T^{2} \)
73 \( 1 - 2.21e6T + 1.10e13T^{2} \)
79 \( 1 + (-1.53e5 - 2.66e5i)T + (-9.60e12 + 1.66e13i)T^{2} \)
83 \( 1 + (-1.57e6 - 2.73e6i)T + (-1.35e13 + 2.35e13i)T^{2} \)
89 \( 1 + 1.93e6T + 4.42e13T^{2} \)
97 \( 1 + (4.94e6 + 8.56e6i)T + (-4.03e13 + 6.99e13i)T^{2} \)
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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

−10.36824367752564272523253906063, −9.356327040465172750448827834696, −8.567892701140852673554375546176, −7.47219039394292785957713788161, −6.70608917045517192358985267945, −5.60681259328841421508018640262, −4.53800704638841454376104426429, −3.50858393941822048441482136494, −2.27410458400551614980469018497, −1.19778765216090330197635413924, 0.35224654186850079528932592827, 1.29853444152019527925291857481, 2.64725956415329620545202859939, 4.03130818628095175308910695510, 4.56787336011291775028435742202, 6.24009658864772658969620096785, 6.54530223906098264858681605863, 8.119171185643135311429516000545, 8.629605486721319186182306064464, 9.535209504086289215387907074247

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