Properties

Label 2-456-1.1-c1-0-2
Degree $2$
Conductor $456$
Sign $1$
Analytic cond. $3.64117$
Root an. cond. $1.90818$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 1.56·5-s − 1.56·7-s + 9-s + 5.56·11-s + 3.12·13-s − 1.56·15-s + 6.68·17-s + 19-s − 1.56·21-s + 9.12·23-s − 2.56·25-s + 27-s − 8.24·29-s − 2·31-s + 5.56·33-s + 2.43·35-s − 8·37-s + 3.12·39-s − 5.12·41-s − 1.56·43-s − 1.56·45-s − 6.68·47-s − 4.56·49-s + 6.68·51-s + 4.24·53-s − 8.68·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.698·5-s − 0.590·7-s + 0.333·9-s + 1.67·11-s + 0.866·13-s − 0.403·15-s + 1.62·17-s + 0.229·19-s − 0.340·21-s + 1.90·23-s − 0.512·25-s + 0.192·27-s − 1.53·29-s − 0.359·31-s + 0.968·33-s + 0.412·35-s − 1.31·37-s + 0.500·39-s − 0.800·41-s − 0.238·43-s − 0.232·45-s − 0.975·47-s − 0.651·49-s + 0.936·51-s + 0.583·53-s − 1.17·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $1$
Analytic conductor: \(3.64117\)
Root analytic conductor: \(1.90818\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 456,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.635141292\)
\(L(\frac12)\) \(\approx\) \(1.635141292\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
19 \( 1 - T \)
good5 \( 1 + 1.56T + 5T^{2} \)
7 \( 1 + 1.56T + 7T^{2} \)
11 \( 1 - 5.56T + 11T^{2} \)
13 \( 1 - 3.12T + 13T^{2} \)
17 \( 1 - 6.68T + 17T^{2} \)
23 \( 1 - 9.12T + 23T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 8T + 37T^{2} \)
41 \( 1 + 5.12T + 41T^{2} \)
43 \( 1 + 1.56T + 43T^{2} \)
47 \( 1 + 6.68T + 47T^{2} \)
53 \( 1 - 4.24T + 53T^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 6.68T + 61T^{2} \)
67 \( 1 - 6.24T + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 16.9T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 4.24T + 97T^{2} \)
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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

−11.24540968226618014154682550052, −9.985716414794461087196564862141, −9.205059440403557799239543917143, −8.470762069674737448807889307182, −7.36924501536222123211794169192, −6.62895753852762574295251684134, −5.35681992211034713205325762601, −3.71097266423840225411789316701, −3.45450757029290982715145021911, −1.35376246294913576804396460802, 1.35376246294913576804396460802, 3.45450757029290982715145021911, 3.71097266423840225411789316701, 5.35681992211034713205325762601, 6.62895753852762574295251684134, 7.36924501536222123211794169192, 8.470762069674737448807889307182, 9.205059440403557799239543917143, 9.985716414794461087196564862141, 11.24540968226618014154682550052

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