Properties

Label 2-84e2-1.1-c1-0-70
Degree $2$
Conductor $7056$
Sign $-1$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
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  − 2·5-s + 4·11-s + 2·13-s + 2·17-s − 4·19-s − 8·23-s − 25-s − 6·29-s + 8·31-s + 6·37-s − 6·41-s − 4·43-s + 2·53-s − 8·55-s − 4·59-s + 2·61-s − 4·65-s + 4·67-s + 8·71-s − 10·73-s + 8·79-s + 4·83-s − 4·85-s − 6·89-s + 8·95-s − 2·97-s − 18·101-s + ⋯
L(s)  = 1  − 0.894·5-s + 1.20·11-s + 0.554·13-s + 0.485·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s − 1.11·29-s + 1.43·31-s + 0.986·37-s − 0.937·41-s − 0.609·43-s + 0.274·53-s − 1.07·55-s − 0.520·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.900·79-s + 0.439·83-s − 0.433·85-s − 0.635·89-s + 0.820·95-s − 0.203·97-s − 1.79·101-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: yes
Arithmetic: yes
Character: $\chi_{7056} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−7.82263580546813636934054242173, −6.78395093605661555810316280628, −6.30966811163732211212778691110, −5.58667709375314371155679753614, −4.44811958782670156328485077225, −3.97597569667552498946447381576, −3.42103471864385914094720232253, −2.20947719434785265756306128762, −1.24950065754749508798543005172, 0, 1.24950065754749508798543005172, 2.20947719434785265756306128762, 3.42103471864385914094720232253, 3.97597569667552498946447381576, 4.44811958782670156328485077225, 5.58667709375314371155679753614, 6.30966811163732211212778691110, 6.78395093605661555810316280628, 7.82263580546813636934054242173

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