Properties

Label 2-7e4-1.1-c3-0-422
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.28·2-s + 8.91·3-s + 10.3·4-s + 18.9·5-s + 38.2·6-s + 10.2·8-s + 52.4·9-s + 81.3·10-s + 2.99·11-s + 92.5·12-s − 36.2·13-s + 169.·15-s − 39.2·16-s + 57.8·17-s + 225.·18-s + 62.8·19-s + 197.·20-s + 12.8·22-s + 15.3·23-s + 91.0·24-s + 235.·25-s − 155.·26-s + 227.·27-s + 44.3·29-s + 725.·30-s − 272.·31-s − 250.·32-s + ⋯
L(s)  = 1  + 1.51·2-s + 1.71·3-s + 1.29·4-s + 1.69·5-s + 2.60·6-s + 0.451·8-s + 1.94·9-s + 2.57·10-s + 0.0819·11-s + 2.22·12-s − 0.772·13-s + 2.91·15-s − 0.613·16-s + 0.825·17-s + 2.94·18-s + 0.758·19-s + 2.20·20-s + 0.124·22-s + 0.139·23-s + 0.774·24-s + 1.88·25-s − 1.17·26-s + 1.61·27-s + 0.283·29-s + 4.41·30-s − 1.57·31-s − 1.38·32-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: \(0\)
Selberg data: \((2,\ 2401,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(14.76268901\)
\(L(\frac12)\) \(\approx\) \(14.76268901\)
\(L(\frac{5}{2})\) 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 \( 1 - 4.28T + 8T^{2} \)
3 \( 1 - 8.91T + 27T^{2} \)
5 \( 1 - 18.9T + 125T^{2} \)
11 \( 1 - 2.99T + 1.33e3T^{2} \)
13 \( 1 + 36.2T + 2.19e3T^{2} \)
17 \( 1 - 57.8T + 4.91e3T^{2} \)
19 \( 1 - 62.8T + 6.85e3T^{2} \)
23 \( 1 - 15.3T + 1.21e4T^{2} \)
29 \( 1 - 44.3T + 2.43e4T^{2} \)
31 \( 1 + 272.T + 2.97e4T^{2} \)
37 \( 1 - 111.T + 5.06e4T^{2} \)
41 \( 1 + 241.T + 6.89e4T^{2} \)
43 \( 1 - 53.4T + 7.95e4T^{2} \)
47 \( 1 - 170.T + 1.03e5T^{2} \)
53 \( 1 - 464.T + 1.48e5T^{2} \)
59 \( 1 + 802.T + 2.05e5T^{2} \)
61 \( 1 + 71.3T + 2.26e5T^{2} \)
67 \( 1 + 223.T + 3.00e5T^{2} \)
71 \( 1 + 153.T + 3.57e5T^{2} \)
73 \( 1 + 935.T + 3.89e5T^{2} \)
79 \( 1 + 279.T + 4.93e5T^{2} \)
83 \( 1 - 273.T + 5.71e5T^{2} \)
89 \( 1 - 1.30e3T + 7.04e5T^{2} \)
97 \( 1 - 1.46e3T + 9.12e5T^{2} \)
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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

−8.848421212942872898706537280930, −7.61553627520695840846769526868, −7.04425534080104835093850027498, −6.03215712108143933416025883605, −5.37704261795392681559270541419, −4.60568484171082495145609845528, −3.54187858700004797523575486322, −2.91286035549626132522643546063, −2.24732577372372223117587357876, −1.48513045488692119166644592980, 1.48513045488692119166644592980, 2.24732577372372223117587357876, 2.91286035549626132522643546063, 3.54187858700004797523575486322, 4.60568484171082495145609845528, 5.37704261795392681559270541419, 6.03215712108143933416025883605, 7.04425534080104835093850027498, 7.61553627520695840846769526868, 8.848421212942872898706537280930

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