Properties

Label 2-28e2-1.1-c3-0-26
Degree $2$
Conductor $784$
Sign $-1$
Analytic cond. $46.2574$
Root an. cond. $6.80128$
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  − 6.08·3-s − 19.1·5-s + 9.99·9-s + 26.5·11-s + 10.3·13-s + 116.·15-s − 101.·17-s + 93.7·19-s + 8.57·23-s + 242.·25-s + 103.·27-s + 52.3·29-s + 55.4·31-s − 161.·33-s + 428.·37-s − 62.8·39-s − 137.·41-s + 172·43-s − 191.·45-s + 49.2·47-s + 616.·51-s − 474.·53-s − 509.·55-s − 570.·57-s − 197.·59-s − 401.·61-s − 198.·65-s + ⋯
L(s)  = 1  − 1.17·3-s − 1.71·5-s + 0.370·9-s + 0.728·11-s + 0.220·13-s + 2.00·15-s − 1.44·17-s + 1.13·19-s + 0.0777·23-s + 1.93·25-s + 0.737·27-s + 0.335·29-s + 0.321·31-s − 0.852·33-s + 1.90·37-s − 0.258·39-s − 0.521·41-s + 0.609·43-s − 0.634·45-s + 0.152·47-s + 1.69·51-s − 1.22·53-s − 1.24·55-s − 1.32·57-s − 0.434·59-s − 0.841·61-s − 0.377·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(46.2574\)
Root analytic conductor: \(6.80128\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 784,\ (\ :3/2),\ -1)\)

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)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + 6.08T + 27T^{2} \)
5 \( 1 + 19.1T + 125T^{2} \)
11 \( 1 - 26.5T + 1.33e3T^{2} \)
13 \( 1 - 10.3T + 2.19e3T^{2} \)
17 \( 1 + 101.T + 4.91e3T^{2} \)
19 \( 1 - 93.7T + 6.85e3T^{2} \)
23 \( 1 - 8.57T + 1.21e4T^{2} \)
29 \( 1 - 52.3T + 2.43e4T^{2} \)
31 \( 1 - 55.4T + 2.97e4T^{2} \)
37 \( 1 - 428.T + 5.06e4T^{2} \)
41 \( 1 + 137.T + 6.89e4T^{2} \)
43 \( 1 - 172T + 7.95e4T^{2} \)
47 \( 1 - 49.2T + 1.03e5T^{2} \)
53 \( 1 + 474.T + 1.48e5T^{2} \)
59 \( 1 + 197.T + 2.05e5T^{2} \)
61 \( 1 + 401.T + 2.26e5T^{2} \)
67 \( 1 + 125.T + 3.00e5T^{2} \)
71 \( 1 + 788.T + 3.57e5T^{2} \)
73 \( 1 - 604.T + 3.89e5T^{2} \)
79 \( 1 + 783.T + 4.93e5T^{2} \)
83 \( 1 - 339.T + 5.71e5T^{2} \)
89 \( 1 - 511.T + 7.04e5T^{2} \)
97 \( 1 + 672.T + 9.12e5T^{2} \)
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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

−9.450424875295049802739073473235, −8.539407360589365635377317724986, −7.65064660121045645297549649191, −6.80992660526513325483475109735, −6.04096884627097917129064406904, −4.76825383697007177605592223196, −4.20416483457064614251192137781, −3.03359448191211918568964883698, −1.00517661319944208336295671969, 0, 1.00517661319944208336295671969, 3.03359448191211918568964883698, 4.20416483457064614251192137781, 4.76825383697007177605592223196, 6.04096884627097917129064406904, 6.80992660526513325483475109735, 7.65064660121045645297549649191, 8.539407360589365635377317724986, 9.450424875295049802739073473235

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