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Properties

Label 2-468-52.51-c0-0-3
Degree $2$
Conductor $468$
Sign $1$
Analytic cond. $0.233562$
Root an. cond. $0.483282$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s − 2·11-s − 13-s + 16-s − 2·22-s + 25-s − 26-s + 32-s − 2·44-s − 2·47-s − 49-s + 50-s − 52-s + 2·59-s − 2·61-s + 64-s + 2·71-s + 2·83-s − 2·88-s − 2·94-s − 98-s + 100-s − 104-s + 2·118-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s − 2·11-s − 13-s + 16-s − 2·22-s + 25-s − 26-s + 32-s − 2·44-s − 2·47-s − 49-s + 50-s − 52-s + 2·59-s − 2·61-s + 64-s + 2·71-s + 2·83-s − 2·88-s − 2·94-s − 98-s + 100-s − 104-s + 2·118-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(468\)    =    \(2^{2} \cdot 3^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(0.233562\)
Root analytic conductor: \(0.483282\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{468} (415, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 468,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.422599735\)
\(L(\frac12)\) \(\approx\) \(1.422599735\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−11.28846195747301840136862002126, −10.53724025155482616035248949595, −9.777035977822234359307775531736, −8.206659658586756605920355131080, −7.50845784289009217052883502173, −6.51247412427874016263825988718, −5.25120987737984177912380869488, −4.80384839605694580965260113375, −3.22421568095521731360694808713, −2.29098313964472879524943890970, 2.29098313964472879524943890970, 3.22421568095521731360694808713, 4.80384839605694580965260113375, 5.25120987737984177912380869488, 6.51247412427874016263825988718, 7.50845784289009217052883502173, 8.206659658586756605920355131080, 9.777035977822234359307775531736, 10.53724025155482616035248949595, 11.28846195747301840136862002126

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