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Properties

Label 2-432-48.35-c1-0-13
Degree $2$
Conductor $432$
Sign $0.611 - 0.790i$
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  + (−1.06 + 0.931i)2-s + (0.264 − 1.98i)4-s + (2.29 + 2.29i)5-s + 3.92·7-s + (1.56 + 2.35i)8-s + (−4.57 − 0.304i)10-s + (1.48 − 1.48i)11-s + (−2.94 − 2.94i)13-s + (−4.17 + 3.65i)14-s + (−3.85 − 1.05i)16-s − 4.04i·17-s + (3.09 − 3.09i)19-s + (5.14 − 3.93i)20-s + (−0.196 + 2.95i)22-s + 6.25i·23-s + ⋯
L(s)  = 1  + (−0.752 + 0.658i)2-s + (0.132 − 0.991i)4-s + (1.02 + 1.02i)5-s + 1.48·7-s + (0.553 + 0.833i)8-s + (−1.44 − 0.0961i)10-s + (0.446 − 0.446i)11-s + (−0.817 − 0.817i)13-s + (−1.11 + 0.977i)14-s + (−0.964 − 0.262i)16-s − 0.980i·17-s + (0.709 − 0.709i)19-s + (1.15 − 0.879i)20-s + (−0.0419 + 0.630i)22-s + 1.30i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.611 - 0.790i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.611 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18879 + 0.583360i\)
\(L(\frac12)\) \(\approx\) \(1.18879 + 0.583360i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.06 - 0.931i)T \)
3 \( 1 \)
good5 \( 1 + (-2.29 - 2.29i)T + 5iT^{2} \)
7 \( 1 - 3.92T + 7T^{2} \)
11 \( 1 + (-1.48 + 1.48i)T - 11iT^{2} \)
13 \( 1 + (2.94 + 2.94i)T + 13iT^{2} \)
17 \( 1 + 4.04iT - 17T^{2} \)
19 \( 1 + (-3.09 + 3.09i)T - 19iT^{2} \)
23 \( 1 - 6.25iT - 23T^{2} \)
29 \( 1 + (1.44 - 1.44i)T - 29iT^{2} \)
31 \( 1 - 8.70iT - 31T^{2} \)
37 \( 1 + (-4.01 + 4.01i)T - 37iT^{2} \)
41 \( 1 + 7.82T + 41T^{2} \)
43 \( 1 + (0.354 + 0.354i)T + 43iT^{2} \)
47 \( 1 + 3.25T + 47T^{2} \)
53 \( 1 + (7.21 + 7.21i)T + 53iT^{2} \)
59 \( 1 + (1.07 - 1.07i)T - 59iT^{2} \)
61 \( 1 + (-8.81 - 8.81i)T + 61iT^{2} \)
67 \( 1 + (4.72 - 4.72i)T - 67iT^{2} \)
71 \( 1 - 9.37iT - 71T^{2} \)
73 \( 1 - 7.02iT - 73T^{2} \)
79 \( 1 + 9.40iT - 79T^{2} \)
83 \( 1 + (4.41 + 4.41i)T + 83iT^{2} \)
89 \( 1 - 4.59T + 89T^{2} \)
97 \( 1 + 19.5T + 97T^{2} \)
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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

−11.14449054214115749173748592098, −10.21484871278124749558261639510, −9.526919455754325639987385361580, −8.527131446761995529847424985757, −7.45522486442943324059724350700, −6.90287077425330680311154484338, −5.56336490158921112964006221538, −5.05718096972729175695714726118, −2.83093340420923104056944719029, −1.50153539986880736705982316200, 1.44857151860355535191047041142, 2.12406890450908456352284007540, 4.22467625690356464700595859389, 4.98012842611535444762020019062, 6.38182065393365633572972847667, 7.77752785886992486051311053618, 8.401260900904827837495469423219, 9.382408239907432386122923196278, 9.907763393814102768066488137731, 10.99542205089783897517002577034

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