Properties

Label 2-7e4-1.1-c3-0-298
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  + 4.01·2-s − 8.06·3-s + 8.08·4-s − 14.0·5-s − 32.3·6-s + 0.326·8-s + 38.0·9-s − 56.3·10-s + 53.6·11-s − 65.1·12-s − 64.5·13-s + 113.·15-s − 63.3·16-s + 52.7·17-s + 152.·18-s + 11.4·19-s − 113.·20-s + 215.·22-s − 98.9·23-s − 2.63·24-s + 72.5·25-s − 258.·26-s − 89.1·27-s + 42.5·29-s + 454.·30-s + 67.2·31-s − 256.·32-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.55·3-s + 1.01·4-s − 1.25·5-s − 2.20·6-s + 0.0144·8-s + 1.40·9-s − 1.78·10-s + 1.47·11-s − 1.56·12-s − 1.37·13-s + 1.95·15-s − 0.989·16-s + 0.753·17-s + 1.99·18-s + 0.137·19-s − 1.26·20-s + 2.08·22-s − 0.896·23-s − 0.0223·24-s + 0.580·25-s − 1.95·26-s − 0.635·27-s + 0.272·29-s + 2.76·30-s + 0.389·31-s − 1.41·32-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 \( 1 - 4.01T + 8T^{2} \)
3 \( 1 + 8.06T + 27T^{2} \)
5 \( 1 + 14.0T + 125T^{2} \)
11 \( 1 - 53.6T + 1.33e3T^{2} \)
13 \( 1 + 64.5T + 2.19e3T^{2} \)
17 \( 1 - 52.7T + 4.91e3T^{2} \)
19 \( 1 - 11.4T + 6.85e3T^{2} \)
23 \( 1 + 98.9T + 1.21e4T^{2} \)
29 \( 1 - 42.5T + 2.43e4T^{2} \)
31 \( 1 - 67.2T + 2.97e4T^{2} \)
37 \( 1 - 428.T + 5.06e4T^{2} \)
41 \( 1 + 45.7T + 6.89e4T^{2} \)
43 \( 1 - 196.T + 7.95e4T^{2} \)
47 \( 1 - 568.T + 1.03e5T^{2} \)
53 \( 1 - 151.T + 1.48e5T^{2} \)
59 \( 1 + 171.T + 2.05e5T^{2} \)
61 \( 1 - 5.02T + 2.26e5T^{2} \)
67 \( 1 + 196.T + 3.00e5T^{2} \)
71 \( 1 - 79.6T + 3.57e5T^{2} \)
73 \( 1 + 719.T + 3.89e5T^{2} \)
79 \( 1 - 241.T + 4.93e5T^{2} \)
83 \( 1 - 421.T + 5.71e5T^{2} \)
89 \( 1 + 628.T + 7.04e5T^{2} \)
97 \( 1 - 1.68e3T + 9.12e5T^{2} \)
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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

−7.78586593157593944402058633648, −7.16673769384919020078092162102, −6.35296587144060600547123433878, −5.80712999657975721876613504039, −4.94324118926201770209397371290, −4.26244011005207298551946282666, −3.86942142653809842715780167831, −2.63667008544041433101574616468, −0.995854883304714586498732627777, 0, 0.995854883304714586498732627777, 2.63667008544041433101574616468, 3.86942142653809842715780167831, 4.26244011005207298551946282666, 4.94324118926201770209397371290, 5.80712999657975721876613504039, 6.35296587144060600547123433878, 7.16673769384919020078092162102, 7.78586593157593944402058633648

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