Properties

Degree $2$
Conductor $7056$
Sign $-0.944 + 0.327i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.87i·5-s + 3.87i·11-s − 3.46i·13-s + 4·19-s + 7.74i·23-s − 10.0·25-s − 6.70·29-s − 31-s + 4·37-s + 7.74i·41-s − 6.92i·43-s − 13.4·47-s + 6.70·53-s − 15.0·55-s − 6.70·59-s + ⋯
L(s)  = 1  + 1.73i·5-s + 1.16i·11-s − 0.960i·13-s + 0.917·19-s + 1.61i·23-s − 2.00·25-s − 1.24·29-s − 0.179·31-s + 0.657·37-s + 1.20i·41-s − 1.05i·43-s − 1.95·47-s + 0.921·53-s − 2.02·55-s − 0.873·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.944 + 0.327i$
Motivic weight: \(1\)
Character: $\chi_{7056} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -0.944 + 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9890785989\)
\(L(\frac12)\) \(\approx\) \(0.9890785989\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.87iT - 5T^{2} \)
11 \( 1 - 3.87iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 7.74iT - 23T^{2} \)
29 \( 1 + 6.70T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 - 7.74iT - 41T^{2} \)
43 \( 1 + 6.92iT - 43T^{2} \)
47 \( 1 + 13.4T + 47T^{2} \)
53 \( 1 - 6.70T + 53T^{2} \)
59 \( 1 + 6.70T + 59T^{2} \)
61 \( 1 - 10.3iT - 61T^{2} \)
67 \( 1 + 6.92iT - 67T^{2} \)
71 \( 1 + 7.74iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 12.1iT - 79T^{2} \)
83 \( 1 + 6.70T + 83T^{2} \)
89 \( 1 - 7.74iT - 89T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
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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

−7.948811663819704835177147092899, −7.47750384734131465577254757325, −7.14940514245963890926321649312, −6.26484611938309083917128957661, −5.63249855836941697766153434085, −4.85201633656257562332589149150, −3.68812297127066742060218331275, −3.26149578250211477175875619857, −2.43026732656221220935317623142, −1.53504394174330797523206353312, 0.24753622179303892227757987686, 1.13749648477910573252393607553, 2.02212308629211435461448464649, 3.20988357544793288953886458600, 4.10755386262384893302972959422, 4.68115378228741512986835231101, 5.42027437219794824652123258076, 5.98687150395457914478296041841, 6.82262999808522883934829963270, 7.75732881414341297770290501540

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