Properties

Label 2-7e4-1.1-c3-0-473
Degree $2$
Conductor $2401$
Sign $-1$
Analytic cond. $141.663$
Root an. cond. $11.9022$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.91·2-s + 6.46·3-s + 16.1·4-s − 16.3·5-s + 31.7·6-s + 40.2·8-s + 14.7·9-s − 80.3·10-s + 35.8·11-s + 104.·12-s − 88.8·13-s − 105.·15-s + 68.4·16-s + 0.0595·17-s + 72.7·18-s − 22.3·19-s − 264.·20-s + 176.·22-s − 72.2·23-s + 260.·24-s + 142.·25-s − 436.·26-s − 78.9·27-s − 123.·29-s − 519.·30-s − 168.·31-s + 14.7·32-s + ⋯
L(s)  = 1  + 1.73·2-s + 1.24·3-s + 2.02·4-s − 1.46·5-s + 2.16·6-s + 1.77·8-s + 0.547·9-s − 2.54·10-s + 0.981·11-s + 2.51·12-s − 1.89·13-s − 1.81·15-s + 1.06·16-s + 0.000849·17-s + 0.952·18-s − 0.270·19-s − 2.95·20-s + 1.70·22-s − 0.654·23-s + 2.21·24-s + 1.13·25-s − 3.29·26-s − 0.562·27-s − 0.788·29-s − 3.16·30-s − 0.978·31-s + 0.0812·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2401\)    =    \(7^{4}\)
Sign: $-1$
Analytic conductor: \(141.663\)
Root analytic conductor: \(11.9022\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2401,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 - 4.91T + 8T^{2} \)
3 \( 1 - 6.46T + 27T^{2} \)
5 \( 1 + 16.3T + 125T^{2} \)
11 \( 1 - 35.8T + 1.33e3T^{2} \)
13 \( 1 + 88.8T + 2.19e3T^{2} \)
17 \( 1 - 0.0595T + 4.91e3T^{2} \)
19 \( 1 + 22.3T + 6.85e3T^{2} \)
23 \( 1 + 72.2T + 1.21e4T^{2} \)
29 \( 1 + 123.T + 2.43e4T^{2} \)
31 \( 1 + 168.T + 2.97e4T^{2} \)
37 \( 1 + 15.3T + 5.06e4T^{2} \)
41 \( 1 + 169.T + 6.89e4T^{2} \)
43 \( 1 - 443.T + 7.95e4T^{2} \)
47 \( 1 + 22.2T + 1.03e5T^{2} \)
53 \( 1 + 221.T + 1.48e5T^{2} \)
59 \( 1 - 194.T + 2.05e5T^{2} \)
61 \( 1 - 438.T + 2.26e5T^{2} \)
67 \( 1 + 469.T + 3.00e5T^{2} \)
71 \( 1 + 862.T + 3.57e5T^{2} \)
73 \( 1 + 64.7T + 3.89e5T^{2} \)
79 \( 1 - 815.T + 4.93e5T^{2} \)
83 \( 1 - 766.T + 5.71e5T^{2} \)
89 \( 1 - 341.T + 7.04e5T^{2} \)
97 \( 1 + 354.T + 9.12e5T^{2} \)
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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

−7.85904220219701500605355519956, −7.43289332456078233077560767457, −6.79196723568215497297150736559, −5.65237013010940405186769694183, −4.68256550034955854368397689912, −4.01780016904936939272789940144, −3.56530244724692132958927270988, −2.71173947923319399370975713406, −1.93282196542916236008534892352, 0, 1.93282196542916236008534892352, 2.71173947923319399370975713406, 3.56530244724692132958927270988, 4.01780016904936939272789940144, 4.68256550034955854368397689912, 5.65237013010940405186769694183, 6.79196723568215497297150736559, 7.43289332456078233077560767457, 7.85904220219701500605355519956

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