Properties

Label 2-28e2-1.1-c5-0-49
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 29.2·3-s + 7.45·5-s + 613.·9-s − 447.·11-s − 383.·13-s − 218.·15-s + 1.87e3·17-s + 7.59·19-s − 2.69e3·23-s − 3.06e3·25-s − 1.08e4·27-s − 8.77e3·29-s + 7.54e3·31-s + 1.31e4·33-s + 6.51e3·37-s + 1.12e4·39-s + 1.79e4·41-s − 1.35e3·43-s + 4.57e3·45-s + 1.82e3·47-s − 5.48e4·51-s + 3.58e4·53-s − 3.33e3·55-s − 222.·57-s − 2.36e4·59-s − 2.70e3·61-s − 2.85e3·65-s + ⋯
L(s)  = 1  − 1.87·3-s + 0.133·5-s + 2.52·9-s − 1.11·11-s − 0.628·13-s − 0.250·15-s + 1.57·17-s + 0.00482·19-s − 1.06·23-s − 0.982·25-s − 2.86·27-s − 1.93·29-s + 1.41·31-s + 2.09·33-s + 0.782·37-s + 1.18·39-s + 1.67·41-s − 0.111·43-s + 0.337·45-s + 0.120·47-s − 2.95·51-s + 1.75·53-s − 0.148·55-s − 0.00905·57-s − 0.883·59-s − 0.0929·61-s − 0.0839·65-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: \(1\)
Selberg data: \((2,\ 784,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 + 29.2T + 243T^{2} \)
5 \( 1 - 7.45T + 3.12e3T^{2} \)
11 \( 1 + 447.T + 1.61e5T^{2} \)
13 \( 1 + 383.T + 3.71e5T^{2} \)
17 \( 1 - 1.87e3T + 1.41e6T^{2} \)
19 \( 1 - 7.59T + 2.47e6T^{2} \)
23 \( 1 + 2.69e3T + 6.43e6T^{2} \)
29 \( 1 + 8.77e3T + 2.05e7T^{2} \)
31 \( 1 - 7.54e3T + 2.86e7T^{2} \)
37 \( 1 - 6.51e3T + 6.93e7T^{2} \)
41 \( 1 - 1.79e4T + 1.15e8T^{2} \)
43 \( 1 + 1.35e3T + 1.47e8T^{2} \)
47 \( 1 - 1.82e3T + 2.29e8T^{2} \)
53 \( 1 - 3.58e4T + 4.18e8T^{2} \)
59 \( 1 + 2.36e4T + 7.14e8T^{2} \)
61 \( 1 + 2.70e3T + 8.44e8T^{2} \)
67 \( 1 - 3.68e4T + 1.35e9T^{2} \)
71 \( 1 + 3.37e4T + 1.80e9T^{2} \)
73 \( 1 + 1.69e4T + 2.07e9T^{2} \)
79 \( 1 - 8.44e4T + 3.07e9T^{2} \)
83 \( 1 - 5.24e4T + 3.93e9T^{2} \)
89 \( 1 - 3.18e4T + 5.58e9T^{2} \)
97 \( 1 + 3.67e4T + 8.58e9T^{2} \)
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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

−9.618694347366780429754166824015, −7.79358340666505403726708502963, −7.45230159646695431408595503068, −6.09080014010807144704541578726, −5.68673931290751840621583355048, −4.90541552089148785401146733833, −3.87844052129769436528324352576, −2.23037751757799508906303591818, −0.917115402185362505151887590598, 0, 0.917115402185362505151887590598, 2.23037751757799508906303591818, 3.87844052129769436528324352576, 4.90541552089148785401146733833, 5.68673931290751840621583355048, 6.09080014010807144704541578726, 7.45230159646695431408595503068, 7.79358340666505403726708502963, 9.618694347366780429754166824015

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