Properties

Label 2-2646-21.20-c1-0-37
Degree $2$
Conductor $2646$
Sign $0.409 + 0.912i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·2-s − 4-s − 0.601·5-s i·8-s − 0.601i·10-s − 1.20i·11-s + 0.649i·13-s + 16-s − 0.564·17-s − 0.249i·19-s + 0.601·20-s + 1.20·22-s + 3.95i·23-s − 4.63·25-s − 0.649·26-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.269·5-s − 0.353i·8-s − 0.190i·10-s − 0.361i·11-s + 0.180i·13-s + 0.250·16-s − 0.136·17-s − 0.0571i·19-s + 0.134·20-s + 0.255·22-s + 0.824i·23-s − 0.927·25-s − 0.127·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 + 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.409 + 0.912i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2645, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.409 + 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7832900346\)
\(L(\frac12)\) \(\approx\) \(0.7832900346\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 0.601T + 5T^{2} \)
11 \( 1 + 1.20iT - 11T^{2} \)
13 \( 1 - 0.649iT - 13T^{2} \)
17 \( 1 + 0.564T + 17T^{2} \)
19 \( 1 + 0.249iT - 19T^{2} \)
23 \( 1 - 3.95iT - 23T^{2} \)
29 \( 1 + 1.37iT - 29T^{2} \)
31 \( 1 + 7.11iT - 31T^{2} \)
37 \( 1 - 5.17T + 37T^{2} \)
41 \( 1 + 7.32T + 41T^{2} \)
43 \( 1 - 9.36T + 43T^{2} \)
47 \( 1 + 9.12T + 47T^{2} \)
53 \( 1 + 10.3iT - 53T^{2} \)
59 \( 1 + 7.26T + 59T^{2} \)
61 \( 1 - 0.865iT - 61T^{2} \)
67 \( 1 + 9.37T + 67T^{2} \)
71 \( 1 + 4.28iT - 71T^{2} \)
73 \( 1 + 15.7iT - 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + 5.58T + 83T^{2} \)
89 \( 1 + 7.62T + 89T^{2} \)
97 \( 1 + 13.2iT - 97T^{2} \)
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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

−8.566341671026086806399104651444, −7.85152528615338282115009856213, −7.34279857037183448923596260361, −6.31619993434465536940882220373, −5.81592262420081146688969146902, −4.83298441433102329558439903154, −4.04022064270080203058972317268, −3.17252457334199468118026339017, −1.82521182315970535529942616180, −0.27309741594948712048088374875, 1.21617275486729990479977364230, 2.35148814812828278959327425184, 3.25144734894054052088377299152, 4.18700955232236654340513364636, 4.86959682045551665131665707811, 5.83056950186492650589873476505, 6.73271415769998659303963512821, 7.62219021824733671879056245365, 8.330580893215037806480885345569, 9.077619051055099370885674326268

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