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Properties

Label 2-468-52.35-c0-0-0
Degree $2$
Conductor $468$
Sign $0.252 + 0.967i$
Analytic cond. $0.233562$
Root an. cond. $0.483282$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + 5-s − 0.999·8-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.499 − 0.866i)20-s + (0.499 + 0.866i)26-s + (−0.5 + 0.866i)29-s + (0.499 + 0.866i)32-s − 0.999·34-s + (0.5 − 0.866i)37-s − 0.999·40-s + (−0.5 + 0.866i)41-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + 5-s − 0.999·8-s + (0.5 − 0.866i)10-s + (−0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + (−0.5 − 0.866i)17-s + (−0.499 − 0.866i)20-s + (0.499 + 0.866i)26-s + (−0.5 + 0.866i)29-s + (0.499 + 0.866i)32-s − 0.999·34-s + (0.5 − 0.866i)37-s − 0.999·40-s + (−0.5 + 0.866i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 468 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $0.252 + 0.967i$
Analytic conductor: \(0.233562\)
Root analytic conductor: \(0.483282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{468} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :0),\ 0.252 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.141111814\)
\(L(\frac12)\) \(\approx\) \(1.141111814\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
13 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 - T + T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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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

−11.17178755249202153026444504164, −10.16774824039036665343499929767, −9.474981459348836110060369362517, −8.838433699421797253616710077852, −7.18901736930110602797385188702, −6.15261192292981051222817989081, −5.22154724644941947220701955620, −4.26230446251324513718013190571, −2.80453757484233681179402955695, −1.76445188992100265516793150510, 2.35071391762943257755708033434, 3.76005911135896979208347662598, 5.03200455844370518760123951268, 5.83620215128545266503005088928, 6.61948930086603037195386193152, 7.72953485535166680375182549934, 8.572433171189068322883356457893, 9.588172811673631742359093579772, 10.34104598934499729089347688508, 11.60651975755106495802025475126

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