Properties

Label 2-414-1.1-c3-0-17
Degree $2$
Conductor $414$
Sign $1$
Analytic cond. $24.4267$
Root an. cond. $4.94234$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·4-s + 13.0·5-s + 18.7·7-s + 8·8-s + 26.1·10-s + 41.1·11-s + 32.6·13-s + 37.4·14-s + 16·16-s − 76.8·17-s − 164.·19-s + 52.2·20-s + 82.2·22-s + 23·23-s + 45.5·25-s + 65.2·26-s + 74.8·28-s − 42.5·29-s + 246.·31-s + 32·32-s − 153.·34-s + 244.·35-s + 281.·37-s − 329.·38-s + 104.·40-s + 330.·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.16·5-s + 1.01·7-s + 0.353·8-s + 0.825·10-s + 1.12·11-s + 0.696·13-s + 0.714·14-s + 0.250·16-s − 1.09·17-s − 1.99·19-s + 0.584·20-s + 0.796·22-s + 0.208·23-s + 0.364·25-s + 0.492·26-s + 0.505·28-s − 0.272·29-s + 1.42·31-s + 0.176·32-s − 0.775·34-s + 1.17·35-s + 1.25·37-s − 1.40·38-s + 0.412·40-s + 1.25·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(24.4267\)
Root analytic conductor: \(4.94234\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.276879431\)
\(L(\frac12)\) \(\approx\) \(4.276879431\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 \)
23 \( 1 - 23T \)
good5 \( 1 - 13.0T + 125T^{2} \)
7 \( 1 - 18.7T + 343T^{2} \)
11 \( 1 - 41.1T + 1.33e3T^{2} \)
13 \( 1 - 32.6T + 2.19e3T^{2} \)
17 \( 1 + 76.8T + 4.91e3T^{2} \)
19 \( 1 + 164.T + 6.85e3T^{2} \)
29 \( 1 + 42.5T + 2.43e4T^{2} \)
31 \( 1 - 246.T + 2.97e4T^{2} \)
37 \( 1 - 281.T + 5.06e4T^{2} \)
41 \( 1 - 330.T + 6.89e4T^{2} \)
43 \( 1 + 160.T + 7.95e4T^{2} \)
47 \( 1 + 578.T + 1.03e5T^{2} \)
53 \( 1 + 621.T + 1.48e5T^{2} \)
59 \( 1 - 347.T + 2.05e5T^{2} \)
61 \( 1 - 372.T + 2.26e5T^{2} \)
67 \( 1 - 884.T + 3.00e5T^{2} \)
71 \( 1 + 777.T + 3.57e5T^{2} \)
73 \( 1 + 188.T + 3.89e5T^{2} \)
79 \( 1 - 200.T + 4.93e5T^{2} \)
83 \( 1 - 49.9T + 5.71e5T^{2} \)
89 \( 1 + 767.T + 7.04e5T^{2} \)
97 \( 1 - 1.62e3T + 9.12e5T^{2} \)
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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

−11.08986853437499792147636598825, −9.994029570591155972997483382277, −8.942865333554473318748254432574, −8.120969140279797155538721709164, −6.46602232825934006633131443330, −6.26191715678287388387211372077, −4.85448109818365890745536899559, −4.08056997116696408655359298016, −2.35921881247030632189357806540, −1.46409029141403172597851334610, 1.46409029141403172597851334610, 2.35921881247030632189357806540, 4.08056997116696408655359298016, 4.85448109818365890745536899559, 6.26191715678287388387211372077, 6.46602232825934006633131443330, 8.120969140279797155538721709164, 8.942865333554473318748254432574, 9.994029570591155972997483382277, 11.08986853437499792147636598825

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