Properties

Label 2-28e2-1.1-c1-0-8
Degree $2$
Conductor $784$
Sign $-1$
Analytic cond. $6.26027$
Root an. cond. $2.50205$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5-s + 6·9-s + 11-s + 2·13-s + 3·15-s + 3·17-s − 5·19-s + 3·23-s − 4·25-s − 9·27-s − 6·29-s + 31-s − 3·33-s − 5·37-s − 6·39-s − 10·41-s + 4·43-s − 6·45-s − 47-s − 9·51-s − 9·53-s − 55-s + 15·57-s − 3·59-s + 3·61-s − 2·65-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s + 2·9-s + 0.301·11-s + 0.554·13-s + 0.774·15-s + 0.727·17-s − 1.14·19-s + 0.625·23-s − 4/5·25-s − 1.73·27-s − 1.11·29-s + 0.179·31-s − 0.522·33-s − 0.821·37-s − 0.960·39-s − 1.56·41-s + 0.609·43-s − 0.894·45-s − 0.145·47-s − 1.26·51-s − 1.23·53-s − 0.134·55-s + 1.98·57-s − 0.390·59-s + 0.384·61-s − 0.248·65-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: yes
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 \( 1 + p T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−10.20620662916625970895775928499, −9.142198290542801682490264433147, −7.998191948773462898561718693054, −7.02672400426876020638909964010, −6.25344459015630410317755022398, −5.50408861827812945615920971927, −4.56048692345506519254695272009, −3.60208456561023604215078140332, −1.52606776117536691060853107483, 0, 1.52606776117536691060853107483, 3.60208456561023604215078140332, 4.56048692345506519254695272009, 5.50408861827812945615920971927, 6.25344459015630410317755022398, 7.02672400426876020638909964010, 7.998191948773462898561718693054, 9.142198290542801682490264433147, 10.20620662916625970895775928499

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