Properties

Degree $2$
Conductor $7056$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s − 3·11-s + 2·13-s − 3·17-s + 19-s + 3·23-s + 4·25-s + 6·29-s + 7·31-s − 37-s − 6·41-s + 4·43-s − 9·47-s − 3·53-s + 9·55-s + 9·59-s − 61-s − 6·65-s + 7·67-s − 73-s + 13·79-s + 12·83-s + 9·85-s − 15·89-s − 3·95-s − 10·97-s − 15·101-s + ⋯
L(s)  = 1  − 1.34·5-s − 0.904·11-s + 0.554·13-s − 0.727·17-s + 0.229·19-s + 0.625·23-s + 4/5·25-s + 1.11·29-s + 1.25·31-s − 0.164·37-s − 0.937·41-s + 0.609·43-s − 1.31·47-s − 0.412·53-s + 1.21·55-s + 1.17·59-s − 0.128·61-s − 0.744·65-s + 0.855·67-s − 0.117·73-s + 1.46·79-s + 1.31·83-s + 0.976·85-s − 1.58·89-s − 0.307·95-s − 1.01·97-s − 1.49·101-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: $-1$
Motivic weight: \(1\)
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 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + T + p T^{2} \)
67 \( 1 - 7 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−7.74413636505362681544679975306, −6.87825343880359698626031038494, −6.41660387874708634947180168153, −5.27459157787197237712169105982, −4.71781315656457505960478098390, −3.96344801521167191005684013673, −3.20685805466682892467401683013, −2.45960729516580556924793991187, −1.07925335739504769647485552259, 0, 1.07925335739504769647485552259, 2.45960729516580556924793991187, 3.20685805466682892467401683013, 3.96344801521167191005684013673, 4.71781315656457505960478098390, 5.27459157787197237712169105982, 6.41660387874708634947180168153, 6.87825343880359698626031038494, 7.74413636505362681544679975306

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