Properties

Label 2-40656-1.1-c1-0-93
Degree $2$
Conductor $40656$
Sign $1$
Analytic cond. $324.639$
Root an. cond. $18.0177$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·5-s − 7-s + 9-s − 6·13-s − 2·15-s − 2·17-s − 4·19-s − 21-s − 8·23-s − 25-s + 27-s + 2·29-s + 2·35-s − 10·37-s − 6·39-s + 6·41-s − 4·43-s − 2·45-s + 49-s − 2·51-s + 6·53-s − 4·57-s − 4·59-s − 6·61-s − 63-s + 12·65-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s − 0.516·15-s − 0.485·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.338·35-s − 1.64·37-s − 0.960·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s + 1/7·49-s − 0.280·51-s + 0.824·53-s − 0.529·57-s − 0.520·59-s − 0.768·61-s − 0.125·63-s + 1.48·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40656\)    =    \(2^{4} \cdot 3 \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(324.639\)
Root analytic conductor: \(18.0177\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((2,\ 40656,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
good5 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.21741830266112, −14.82547552239938, −14.32073778137976, −13.76012115506008, −13.27977199582863, −12.50602565997557, −12.19268809611353, −11.88434003325277, −11.11123639806556, −10.42404219814767, −10.02301718334159, −9.550477342609372, −8.731896428011516, −8.487617427764268, −7.635358040065081, −7.469528998064449, −6.776737043418665, −6.167322445720057, −5.416361705336597, −4.591619025574454, −4.213284036471912, −3.648303230347147, −2.825378824637252, −2.310236779608618, −1.575467768663380, 0, 0, 1.575467768663380, 2.310236779608618, 2.825378824637252, 3.648303230347147, 4.213284036471912, 4.591619025574454, 5.416361705336597, 6.167322445720057, 6.776737043418665, 7.469528998064449, 7.635358040065081, 8.487617427764268, 8.731896428011516, 9.550477342609372, 10.02301718334159, 10.42404219814767, 11.11123639806556, 11.88434003325277, 12.19268809611353, 12.50602565997557, 13.27977199582863, 13.76012115506008, 14.32073778137976, 14.82547552239938, 15.21741830266112

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