Properties

Label 2-84e2-1.1-c1-0-60
Degree $2$
Conductor $7056$
Sign $-1$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82·5-s − 2·11-s + 1.41·17-s + 7.07·19-s − 4·23-s + 3.00·25-s − 2·29-s − 8.48·31-s + 10·37-s + 9.89·41-s − 2·43-s + 2.82·47-s + 2·53-s + 5.65·55-s − 1.41·59-s + 2.82·61-s − 12·67-s − 12·71-s − 1.41·73-s + 4·79-s + 9.89·83-s − 4.00·85-s + 7.07·89-s − 20.0·95-s + 9.89·97-s − 8.48·101-s − 2.82·103-s + ⋯
L(s)  = 1  − 1.26·5-s − 0.603·11-s + 0.342·17-s + 1.62·19-s − 0.834·23-s + 0.600·25-s − 0.371·29-s − 1.52·31-s + 1.64·37-s + 1.54·41-s − 0.304·43-s + 0.412·47-s + 0.274·53-s + 0.762·55-s − 0.184·59-s + 0.362·61-s − 1.46·67-s − 1.42·71-s − 0.165·73-s + 0.450·79-s + 1.08·83-s − 0.433·85-s + 0.749·89-s − 2.05·95-s + 1.00·97-s − 0.844·101-s − 0.278·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2.82T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 1.41T + 17T^{2} \)
19 \( 1 - 7.07T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 8.48T + 31T^{2} \)
37 \( 1 - 10T + 37T^{2} \)
41 \( 1 - 9.89T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 2T + 53T^{2} \)
59 \( 1 + 1.41T + 59T^{2} \)
61 \( 1 - 2.82T + 61T^{2} \)
67 \( 1 + 12T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + 1.41T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 9.89T + 83T^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 - 9.89T + 97T^{2} \)
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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

−7.63419421206520839938148304363, −7.26829397609381102579143533808, −6.07228874543160584398886233676, −5.50747150414252939006727864963, −4.64617879179855531935761468529, −3.90227145086373314331756027215, −3.29573261595275243247692281225, −2.40265991502594411286578241085, −1.10626012370763942716280699843, 0, 1.10626012370763942716280699843, 2.40265991502594411286578241085, 3.29573261595275243247692281225, 3.90227145086373314331756027215, 4.64617879179855531935761468529, 5.50747150414252939006727864963, 6.07228874543160584398886233676, 7.26829397609381102579143533808, 7.63419421206520839938148304363

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