Properties

Label 2-28e2-1.1-c5-0-72
Degree $2$
Conductor $784$
Sign $1$
Analytic cond. $125.740$
Root an. cond. $11.2134$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 23.5·3-s + 74.2·5-s + 310.·9-s + 424.·11-s + 252.·13-s + 1.74e3·15-s + 1.10e3·17-s − 6.47·19-s + 3.61e3·23-s + 2.39e3·25-s + 1.57e3·27-s − 5.00e3·29-s + 2.82e3·31-s + 9.97e3·33-s − 2.04e3·37-s + 5.93e3·39-s − 9.39e3·41-s − 1.03e4·43-s + 2.30e4·45-s + 1.70e4·47-s + 2.59e4·51-s − 3.95e4·53-s + 3.15e4·55-s − 152.·57-s + 3.39e4·59-s − 2.82e4·61-s + 1.87e4·65-s + ⋯
L(s)  = 1  + 1.50·3-s + 1.32·5-s + 1.27·9-s + 1.05·11-s + 0.413·13-s + 2.00·15-s + 0.926·17-s − 0.00411·19-s + 1.42·23-s + 0.765·25-s + 0.416·27-s − 1.10·29-s + 0.527·31-s + 1.59·33-s − 0.245·37-s + 0.624·39-s − 0.872·41-s − 0.851·43-s + 1.69·45-s + 1.12·47-s + 1.39·51-s − 1.93·53-s + 1.40·55-s − 0.00620·57-s + 1.26·59-s − 0.973·61-s + 0.550·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(125.740\)
Root analytic conductor: \(11.2134\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(6.353811252\)
\(L(\frac12)\) \(\approx\) \(6.353811252\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 - 23.5T + 243T^{2} \)
5 \( 1 - 74.2T + 3.12e3T^{2} \)
11 \( 1 - 424.T + 1.61e5T^{2} \)
13 \( 1 - 252.T + 3.71e5T^{2} \)
17 \( 1 - 1.10e3T + 1.41e6T^{2} \)
19 \( 1 + 6.47T + 2.47e6T^{2} \)
23 \( 1 - 3.61e3T + 6.43e6T^{2} \)
29 \( 1 + 5.00e3T + 2.05e7T^{2} \)
31 \( 1 - 2.82e3T + 2.86e7T^{2} \)
37 \( 1 + 2.04e3T + 6.93e7T^{2} \)
41 \( 1 + 9.39e3T + 1.15e8T^{2} \)
43 \( 1 + 1.03e4T + 1.47e8T^{2} \)
47 \( 1 - 1.70e4T + 2.29e8T^{2} \)
53 \( 1 + 3.95e4T + 4.18e8T^{2} \)
59 \( 1 - 3.39e4T + 7.14e8T^{2} \)
61 \( 1 + 2.82e4T + 8.44e8T^{2} \)
67 \( 1 + 5.61e4T + 1.35e9T^{2} \)
71 \( 1 - 1.55e4T + 1.80e9T^{2} \)
73 \( 1 - 7.82e4T + 2.07e9T^{2} \)
79 \( 1 - 4.53e4T + 3.07e9T^{2} \)
83 \( 1 - 1.38e3T + 3.93e9T^{2} \)
89 \( 1 - 6.88e4T + 5.58e9T^{2} \)
97 \( 1 + 1.08e5T + 8.58e9T^{2} \)
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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

−9.355372912020021730241351118574, −8.931459943132264806971307981586, −8.002199276282416304442161762037, −7.00402485784650260756494603427, −6.12633707900141863610938475649, −5.08013532794258773703006378565, −3.74072775402439171627684324660, −3.00390087799909120649706369520, −1.91865051537295262688619234919, −1.21947282295432763668721103644, 1.21947282295432763668721103644, 1.91865051537295262688619234919, 3.00390087799909120649706369520, 3.74072775402439171627684324660, 5.08013532794258773703006378565, 6.12633707900141863610938475649, 7.00402485784650260756494603427, 8.002199276282416304442161762037, 8.931459943132264806971307981586, 9.355372912020021730241351118574

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