Properties

Label 2-414-1.1-c7-0-7
Degree $2$
Conductor $414$
Sign $1$
Analytic cond. $129.327$
Root an. cond. $11.3722$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s + 64·4-s − 340.·5-s + 1.26e3·7-s − 512·8-s + 2.72e3·10-s − 5.10e3·11-s − 5.81e3·13-s − 1.01e4·14-s + 4.09e3·16-s − 6.40e3·17-s − 3.85e4·19-s − 2.18e4·20-s + 4.08e4·22-s + 1.21e4·23-s + 3.81e4·25-s + 4.65e4·26-s + 8.10e4·28-s + 2.08e5·29-s + 3.65e3·31-s − 3.27e4·32-s + 5.12e4·34-s − 4.32e5·35-s + 2.39e4·37-s + 3.08e5·38-s + 1.74e5·40-s + 1.15e3·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.21·5-s + 1.39·7-s − 0.353·8-s + 0.862·10-s − 1.15·11-s − 0.733·13-s − 0.987·14-s + 0.250·16-s − 0.316·17-s − 1.28·19-s − 0.609·20-s + 0.817·22-s + 0.208·23-s + 0.488·25-s + 0.519·26-s + 0.698·28-s + 1.58·29-s + 0.0220·31-s − 0.176·32-s + 0.223·34-s − 1.70·35-s + 0.0776·37-s + 0.912·38-s + 0.431·40-s + 0.00260·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(414\)    =    \(2 \cdot 3^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(129.327\)
Root analytic conductor: \(11.3722\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 414,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.6734452371\)
\(L(\frac12)\) \(\approx\) \(0.6734452371\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 8T \)
3 \( 1 \)
23 \( 1 - 1.21e4T \)
good5 \( 1 + 340.T + 7.81e4T^{2} \)
7 \( 1 - 1.26e3T + 8.23e5T^{2} \)
11 \( 1 + 5.10e3T + 1.94e7T^{2} \)
13 \( 1 + 5.81e3T + 6.27e7T^{2} \)
17 \( 1 + 6.40e3T + 4.10e8T^{2} \)
19 \( 1 + 3.85e4T + 8.93e8T^{2} \)
29 \( 1 - 2.08e5T + 1.72e10T^{2} \)
31 \( 1 - 3.65e3T + 2.75e10T^{2} \)
37 \( 1 - 2.39e4T + 9.49e10T^{2} \)
41 \( 1 - 1.15e3T + 1.94e11T^{2} \)
43 \( 1 + 4.89e5T + 2.71e11T^{2} \)
47 \( 1 + 9.88e5T + 5.06e11T^{2} \)
53 \( 1 - 7.77e5T + 1.17e12T^{2} \)
59 \( 1 + 1.61e6T + 2.48e12T^{2} \)
61 \( 1 + 1.04e6T + 3.14e12T^{2} \)
67 \( 1 + 2.06e6T + 6.06e12T^{2} \)
71 \( 1 + 3.58e6T + 9.09e12T^{2} \)
73 \( 1 + 9.75e5T + 1.10e13T^{2} \)
79 \( 1 - 6.00e6T + 1.92e13T^{2} \)
83 \( 1 - 5.40e6T + 2.71e13T^{2} \)
89 \( 1 - 4.13e6T + 4.42e13T^{2} \)
97 \( 1 - 1.18e7T + 8.07e13T^{2} \)
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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

−10.28476715094831008635802485870, −8.840482129876323589997560068762, −8.041407965435592239390344740521, −7.74313082165342555468285113204, −6.60134879928259948773033481436, −5.04820575153970620970453763475, −4.38518405696720448800862317563, −2.87595640919009674400180911102, −1.79445984352462940960931523897, −0.40773603827566765933088438751, 0.40773603827566765933088438751, 1.79445984352462940960931523897, 2.87595640919009674400180911102, 4.38518405696720448800862317563, 5.04820575153970620970453763475, 6.60134879928259948773033481436, 7.74313082165342555468285113204, 8.041407965435592239390344740521, 8.840482129876323589997560068762, 10.28476715094831008635802485870

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