Properties

Label 2-28e2-1.1-c3-0-23
Degree $2$
Conductor $784$
Sign $1$
Analytic cond. $46.2574$
Root an. cond. $6.80128$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·3-s − 7·5-s + 22·9-s + 5·11-s + 14·13-s − 49·15-s + 21·17-s + 49·19-s + 159·23-s − 76·25-s − 35·27-s + 58·29-s + 147·31-s + 35·33-s + 219·37-s + 98·39-s − 350·41-s + 124·43-s − 154·45-s + 525·47-s + 147·51-s + 303·53-s − 35·55-s + 343·57-s − 105·59-s + 413·61-s − 98·65-s + ⋯
L(s)  = 1  + 1.34·3-s − 0.626·5-s + 0.814·9-s + 0.137·11-s + 0.298·13-s − 0.843·15-s + 0.299·17-s + 0.591·19-s + 1.44·23-s − 0.607·25-s − 0.249·27-s + 0.371·29-s + 0.851·31-s + 0.184·33-s + 0.973·37-s + 0.402·39-s − 1.33·41-s + 0.439·43-s − 0.510·45-s + 1.62·47-s + 0.403·51-s + 0.785·53-s − 0.0858·55-s + 0.797·57-s − 0.231·59-s + 0.866·61-s − 0.187·65-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: yes
Arithmetic: yes
Character: $\chi_{784} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 784,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.162846007\)
\(L(\frac12)\) \(\approx\) \(3.162846007\)
\(L(\frac{5}{2})\) 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 \( 1 - 7 T + p^{3} T^{2} \)
5 \( 1 + 7 T + p^{3} T^{2} \)
11 \( 1 - 5 T + p^{3} T^{2} \)
13 \( 1 - 14 T + p^{3} T^{2} \)
17 \( 1 - 21 T + p^{3} T^{2} \)
19 \( 1 - 49 T + p^{3} T^{2} \)
23 \( 1 - 159 T + p^{3} T^{2} \)
29 \( 1 - 2 p T + p^{3} T^{2} \)
31 \( 1 - 147 T + p^{3} T^{2} \)
37 \( 1 - 219 T + p^{3} T^{2} \)
41 \( 1 + 350 T + p^{3} T^{2} \)
43 \( 1 - 124 T + p^{3} T^{2} \)
47 \( 1 - 525 T + p^{3} T^{2} \)
53 \( 1 - 303 T + p^{3} T^{2} \)
59 \( 1 + 105 T + p^{3} T^{2} \)
61 \( 1 - 413 T + p^{3} T^{2} \)
67 \( 1 + 415 T + p^{3} T^{2} \)
71 \( 1 - 432 T + p^{3} T^{2} \)
73 \( 1 - 1113 T + p^{3} T^{2} \)
79 \( 1 - 103 T + p^{3} T^{2} \)
83 \( 1 - 1092 T + p^{3} T^{2} \)
89 \( 1 - 329 T + p^{3} T^{2} \)
97 \( 1 - 882 T + p^{3} T^{2} \)
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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

−9.639011571948089677396583884510, −8.985610332450537137496765523281, −8.179698078487194825445376135982, −7.60121792556273305149321057935, −6.65979864420003067104310498851, −5.33452763140835086955851225117, −4.12582369288083931118925976728, −3.33873277186495172617789518009, −2.44369472001224923099194899957, −0.975314630594255098773091383016, 0.975314630594255098773091383016, 2.44369472001224923099194899957, 3.33873277186495172617789518009, 4.12582369288083931118925976728, 5.33452763140835086955851225117, 6.65979864420003067104310498851, 7.60121792556273305149321057935, 8.179698078487194825445376135982, 8.985610332450537137496765523281, 9.639011571948089677396583884510

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