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Properties

Label 2-56e2-1.1-c1-0-76
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 $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s + 9-s − 4·11-s − 6·13-s + 4·15-s − 4·17-s − 6·19-s − 4·23-s − 25-s + 4·27-s + 6·29-s − 4·31-s + 8·33-s + 6·37-s + 12·39-s + 4·41-s − 12·43-s − 2·45-s − 12·47-s + 8·51-s − 6·53-s + 8·55-s + 12·57-s − 6·59-s + 6·61-s + 12·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 1.03·15-s − 0.970·17-s − 1.37·19-s − 0.834·23-s − 1/5·25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s + 1.39·33-s + 0.986·37-s + 1.92·39-s + 0.624·41-s − 1.82·43-s − 0.298·45-s − 1.75·47-s + 1.12·51-s − 0.824·53-s + 1.07·55-s + 1.58·57-s − 0.781·59-s + 0.768·61-s + 1.48·65-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: \(2\)
Selberg data: \((2,\ 3136,\ (\ :1/2),\ 1)\)

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 \( 1 + 2 T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−8.010034901259649465685255036535, −7.08238367024135924621899684128, −6.45217559209488516158049772833, −5.59673699856690350652282502514, −4.69839552641980880408014499587, −4.43201465917406582204510331599, −3.00626769255360649662431678291, −2.08688930206937100638100581615, 0, 0, 2.08688930206937100638100581615, 3.00626769255360649662431678291, 4.43201465917406582204510331599, 4.69839552641980880408014499587, 5.59673699856690350652282502514, 6.45217559209488516158049772833, 7.08238367024135924621899684128, 8.010034901259649465685255036535

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