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Properties

Label 2-464-29.12-c2-0-11
Degree $2$
Conductor $464$
Sign $-0.200 - 0.979i$
Analytic cond. $12.6430$
Root an. cond. $3.55571$
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  + (3.14 + 3.14i)3-s + 2.76i·5-s + 4.43·7-s + 10.7i·9-s + (0.207 + 0.207i)11-s + 11.2i·13-s + (−8.69 + 8.69i)15-s + (8.91 + 8.91i)17-s + (−7.81 − 7.81i)19-s + (13.9 + 13.9i)21-s − 9.38·23-s + 17.3·25-s + (−5.49 + 5.49i)27-s + (0.343 + 28.9i)29-s + (−18.3 − 18.3i)31-s + ⋯
L(s)  = 1  + (1.04 + 1.04i)3-s + 0.553i·5-s + 0.633·7-s + 1.19i·9-s + (0.0188 + 0.0188i)11-s + 0.866i·13-s + (−0.579 + 0.579i)15-s + (0.524 + 0.524i)17-s + (−0.411 − 0.411i)19-s + (0.663 + 0.663i)21-s − 0.408·23-s + 0.693·25-s + (−0.203 + 0.203i)27-s + (0.0118 + 0.999i)29-s + (−0.591 − 0.591i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.200 - 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $-0.200 - 0.979i$
Analytic conductor: \(12.6430\)
Root analytic conductor: \(3.55571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (273, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1),\ -0.200 - 0.979i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.560357772\)
\(L(\frac12)\) \(\approx\) \(2.560357772\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (-0.343 - 28.9i)T \)
good3 \( 1 + (-3.14 - 3.14i)T + 9iT^{2} \)
5 \( 1 - 2.76iT - 25T^{2} \)
7 \( 1 - 4.43T + 49T^{2} \)
11 \( 1 + (-0.207 - 0.207i)T + 121iT^{2} \)
13 \( 1 - 11.2iT - 169T^{2} \)
17 \( 1 + (-8.91 - 8.91i)T + 289iT^{2} \)
19 \( 1 + (7.81 + 7.81i)T + 361iT^{2} \)
23 \( 1 + 9.38T + 529T^{2} \)
31 \( 1 + (18.3 + 18.3i)T + 961iT^{2} \)
37 \( 1 + (25.0 - 25.0i)T - 1.36e3iT^{2} \)
41 \( 1 + (-20.8 + 20.8i)T - 1.68e3iT^{2} \)
43 \( 1 + (-0.636 - 0.636i)T + 1.84e3iT^{2} \)
47 \( 1 + (-41.9 + 41.9i)T - 2.20e3iT^{2} \)
53 \( 1 + 83.3T + 2.80e3T^{2} \)
59 \( 1 - 64.5T + 3.48e3T^{2} \)
61 \( 1 + (-2.97 - 2.97i)T + 3.72e3iT^{2} \)
67 \( 1 - 6.49iT - 4.48e3T^{2} \)
71 \( 1 + 6.84iT - 5.04e3T^{2} \)
73 \( 1 + (-32.2 + 32.2i)T - 5.32e3iT^{2} \)
79 \( 1 + (63.7 + 63.7i)T + 6.24e3iT^{2} \)
83 \( 1 - 0.509T + 6.88e3T^{2} \)
89 \( 1 + (-89.8 - 89.8i)T + 7.92e3iT^{2} \)
97 \( 1 + (-87.1 + 87.1i)T - 9.40e3iT^{2} \)
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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

−10.81353745420063051296004728050, −10.21027584841908726130826591538, −9.197402703908582148910884613199, −8.601717355092605248608559100032, −7.62930749464774344032981531544, −6.53482868765809641669280761607, −5.08861561167929202173945941792, −4.12689702036648753665458320397, −3.21683972283881298548984086188, −1.97531393682225683115966041546, 0.992813516396229995535278304035, 2.16892904890099854516587576336, 3.35431613442216073261155303190, 4.78748920291734901176109054277, 5.94088694374435679528388145164, 7.22411686738488680349771947007, 7.944868793944603991553015093164, 8.510779268035516561040182855091, 9.422143286619070096459684466057, 10.55740581938465775416248749020

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