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Properties

Label 2-2646-9.4-c1-0-10
Degree $2$
Conductor $2646$
Sign $0.945 - 0.326i$
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  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.59 − 2.75i)5-s + 0.999·8-s − 3.18·10-s + (1.59 + 2.75i)11-s + (−2.85 + 4.93i)13-s + (−0.5 − 0.866i)16-s + 1.52·17-s − 1.28·19-s + (1.59 + 2.75i)20-s + (1.59 − 2.75i)22-s + (1.11 − 1.93i)23-s + (−2.56 − 4.43i)25-s + 5.70·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.711 − 1.23i)5-s + 0.353·8-s − 1.00·10-s + (0.479 + 0.830i)11-s + (−0.790 + 1.36i)13-s + (−0.125 − 0.216i)16-s + 0.369·17-s − 0.294·19-s + (0.355 + 0.616i)20-s + (0.339 − 0.587i)22-s + (0.233 − 0.404i)23-s + (−0.512 − 0.887i)25-s + 1.11·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.945 - 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.945 - 0.326i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1765, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.945 - 0.326i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.358407795\)
\(L(\frac12)\) \(\approx\) \(1.358407795\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1.59 + 2.75i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.59 - 2.75i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.85 - 4.93i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.52T + 17T^{2} \)
19 \( 1 + 1.28T + 19T^{2} \)
23 \( 1 + (-1.11 + 1.93i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.54 - 6.13i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.71 - 8.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (2.80 - 4.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.41 - 5.91i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.91 - 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.05T + 53T^{2} \)
59 \( 1 + (-0.562 + 0.974i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.56 - 2.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.48 - 9.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.69T + 71T^{2} \)
73 \( 1 + 4.96T + 73T^{2} \)
79 \( 1 + (-2.06 - 3.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.03 + 6.98i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 0.225T + 89T^{2} \)
97 \( 1 + (7.42 + 12.8i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.847642076484371269113168855635, −8.668967898852867681379005079350, −7.34460603981298886010161309341, −6.78837840855024876932785292093, −5.65267278532464066046171637838, −4.65676320103348281024123395156, −4.42520291332743537349855257992, −2.99855161242341093908000291839, −1.83355682016759914144432080833, −1.29067673142139025170098219401, 0.51816314391958849534831881699, 2.13335514978875195336900468643, 3.00248802034643766666569589481, 3.94787270898438915038595945621, 5.33894924773692192124641037683, 5.83399691191231854513087522649, 6.50556746316720168294879759474, 7.34492825298989952097937540102, 7.87244513829951634589069705798, 8.806768374012014341997681806867

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