Properties

Label 2-56e2-1.1-c1-0-3
Degree $2$
Conductor $3136$
Sign $1$
Analytic cond. $25.0410$
Root an. cond. $5.00410$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 2·9-s − 3·11-s − 6·13-s + 15-s + 5·17-s + 19-s − 7·23-s − 4·25-s + 5·27-s − 2·29-s + 5·31-s + 3·33-s − 3·37-s + 6·39-s + 2·41-s + 4·43-s + 2·45-s − 5·47-s − 5·51-s + 53-s + 3·55-s − 57-s + 15·59-s − 5·61-s + 6·65-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.904·11-s − 1.66·13-s + 0.258·15-s + 1.21·17-s + 0.229·19-s − 1.45·23-s − 4/5·25-s + 0.962·27-s − 0.371·29-s + 0.898·31-s + 0.522·33-s − 0.493·37-s + 0.960·39-s + 0.312·41-s + 0.609·43-s + 0.298·45-s − 0.729·47-s − 0.700·51-s + 0.137·53-s + 0.404·55-s − 0.132·57-s + 1.95·59-s − 0.640·61-s + 0.744·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(25.0410\)
Root analytic conductor: \(5.00410\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6379074933\)
\(L(\frac12)\) \(\approx\) \(0.6379074933\)
\(L(\frac{3}{2})\) 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 \( 1 + T + p T^{2} \)
5 \( 1 + T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 5 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 15 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 - T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 7 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−8.477656549727874393422066847084, −7.82830516711369004018271381451, −7.39447814586933003973621730528, −6.31319457642689362922902445523, −5.50124088065579204535865736120, −5.06511071409866508052902087436, −4.04159955816643176035584796125, −3.00645085864571493614343512088, −2.17025746450353686276979292547, −0.46739477050181469477720707475, 0.46739477050181469477720707475, 2.17025746450353686276979292547, 3.00645085864571493614343512088, 4.04159955816643176035584796125, 5.06511071409866508052902087436, 5.50124088065579204535865736120, 6.31319457642689362922902445523, 7.39447814586933003973621730528, 7.82830516711369004018271381451, 8.477656549727874393422066847084

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