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Properties

Label 2-56e2-1.1-c1-0-45
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 9-s + 4·11-s − 4·13-s + 2·17-s − 6·19-s + 8·23-s − 5·25-s + 4·27-s − 2·29-s + 4·31-s − 8·33-s − 10·37-s + 8·39-s + 10·41-s − 4·43-s − 4·47-s − 4·51-s + 2·53-s + 12·57-s + 10·59-s − 8·61-s + 8·67-s − 16·69-s + 6·73-s + 10·75-s − 16·79-s + ⋯
L(s)  = 1  − 1.15·3-s + 1/3·9-s + 1.20·11-s − 1.10·13-s + 0.485·17-s − 1.37·19-s + 1.66·23-s − 25-s + 0.769·27-s − 0.371·29-s + 0.718·31-s − 1.39·33-s − 1.64·37-s + 1.28·39-s + 1.56·41-s − 0.609·43-s − 0.583·47-s − 0.560·51-s + 0.274·53-s + 1.58·57-s + 1.30·59-s − 1.02·61-s + 0.977·67-s − 1.92·69-s + 0.702·73-s + 1.15·75-s − 1.80·79-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: \(1\)
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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
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

−8.393062466480562650469160193844, −7.29018180537558040819941052896, −6.73348480631114685674705672568, −6.06219886229881942361734043909, −5.26038869900191935705537529193, −4.59785854498247297237295249330, −3.69866614710771098862034755555, −2.49712879819427106736009257667, −1.25930884241969740273107021321, 0, 1.25930884241969740273107021321, 2.49712879819427106736009257667, 3.69866614710771098862034755555, 4.59785854498247297237295249330, 5.26038869900191935705537529193, 6.06219886229881942361734043909, 6.73348480631114685674705672568, 7.29018180537558040819941052896, 8.393062466480562650469160193844

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