Properties

Label 2-28e2-1.1-c1-0-16
Degree $2$
Conductor $784$
Sign $-1$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·3-s − 2.82·5-s − 0.999·9-s − 6·11-s + 5.65·13-s − 4.00·15-s − 1.41·17-s − 4.24·19-s − 4·23-s + 3.00·25-s − 5.65·27-s − 6·29-s + 2.82·31-s − 8.48·33-s + 2·37-s + 8.00·39-s + 1.41·41-s − 10·43-s + 2.82·45-s − 2.82·47-s − 2.00·51-s − 2·53-s + 16.9·55-s − 6·57-s + 1.41·59-s + 8.48·61-s − 16.0·65-s + ⋯
L(s)  = 1  + 0.816·3-s − 1.26·5-s − 0.333·9-s − 1.80·11-s + 1.56·13-s − 1.03·15-s − 0.342·17-s − 0.973·19-s − 0.834·23-s + 0.600·25-s − 1.08·27-s − 1.11·29-s + 0.508·31-s − 1.47·33-s + 0.328·37-s + 1.28·39-s + 0.220·41-s − 1.52·43-s + 0.421·45-s − 0.412·47-s − 0.280·51-s − 0.274·53-s + 2.28·55-s − 0.794·57-s + 0.184·59-s + 1.08·61-s − 1.98·65-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(784\)    =    \(2^{4} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(6.26027\)
Root analytic conductor: \(2.50205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{784} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 784,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41T + 3T^{2} \)
5 \( 1 + 2.82T + 5T^{2} \)
11 \( 1 + 6T + 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 + 1.41T + 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 2.82T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 - 1.41T + 41T^{2} \)
43 \( 1 + 10T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 1.41T + 59T^{2} \)
61 \( 1 - 8.48T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 9.89T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 1.41T + 83T^{2} \)
89 \( 1 - 4.24T + 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−9.845809419307233483499829762170, −8.608238699688466466833484896727, −8.204114441276929531445016516329, −7.70842157424562848230327746919, −6.42822498836810409780446243939, −5.33360296629171073414900523560, −4.07090503218090234482268790212, −3.36435802030019752953020125633, −2.23667764518565054470265041151, 0, 2.23667764518565054470265041151, 3.36435802030019752953020125633, 4.07090503218090234482268790212, 5.33360296629171073414900523560, 6.42822498836810409780446243939, 7.70842157424562848230327746919, 8.204114441276929531445016516329, 8.608238699688466466833484896727, 9.845809419307233483499829762170

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