Properties

Label 2-2646-63.16-c1-0-21
Degree $2$
Conductor $2646$
Sign $0.282 + 0.959i$
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 + 1.93·5-s + 0.999·8-s + (−0.965 − 1.67i)10-s − 5.46·11-s + (−1.22 − 2.12i)13-s + (−0.5 − 0.866i)16-s + (3.15 + 5.46i)17-s + (0.965 − 1.67i)19-s + (−0.965 + 1.67i)20-s + (2.73 + 4.73i)22-s + 5.92·23-s − 1.26·25-s + (−1.22 + 2.12i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + 0.863·5-s + 0.353·8-s + (−0.305 − 0.529i)10-s − 1.64·11-s + (−0.339 − 0.588i)13-s + (−0.125 − 0.216i)16-s + (0.765 + 1.32i)17-s + (0.221 − 0.383i)19-s + (−0.215 + 0.374i)20-s + (0.582 + 1.00i)22-s + 1.23·23-s − 0.253·25-s + (−0.240 + 0.416i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.282 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.282 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.282 + 0.959i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.282 + 0.959i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.476617356\)
\(L(\frac12)\) \(\approx\) \(1.476617356\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.93T + 5T^{2} \)
11 \( 1 + 5.46T + 11T^{2} \)
13 \( 1 + (1.22 + 2.12i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.15 - 5.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.965 + 1.67i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.92T + 23T^{2} \)
29 \( 1 + (0.366 - 0.633i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.67 + 6.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4 + 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.82 - 4.89i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.901 + 1.56i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.76 + 8.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.63 - 2.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.50 + 4.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.48 - 2.56i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.09 + 12.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 + (4.76 + 8.24i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.06 - 8.76i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.94 - 8.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.64 + 11.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.93 - 3.34i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.768359499905785153878795485214, −7.947106496098909934991650316439, −7.52075494048384914605857367183, −6.27977567777407029087557596150, −5.50034466067140765176563636686, −4.90073672992394243228097606418, −3.64236535534424959546552834667, −2.70502759992909340590515234749, −2.02947415173125384913812430553, −0.65826361970598559773699188475, 0.990919243859700706461767446834, 2.33432580120155383197758646530, 3.11508479549009132463463785224, 4.71906913233468051347682912624, 5.20020095701611029994095280304, 5.88932649046936872448190125638, 6.84605150032251342564558528801, 7.47610039790678192284039186935, 8.173342287437699033788932693961, 9.031645341483326806533598061685

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