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Properties

Label 2-2646-63.41-c1-0-5
Degree $2$
Conductor $2646$
Sign $-0.147 - 0.988i$
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  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.220 + 0.381i)5-s + 0.999i·8-s − 0.440i·10-s + (0.450 − 0.260i)11-s + (−5.55 − 3.20i)13-s + (−0.5 − 0.866i)16-s − 0.327·17-s − 4.23i·19-s + (0.220 + 0.381i)20-s + (−0.260 + 0.450i)22-s + (−1.25 − 0.725i)23-s + (2.40 + 4.16i)25-s + 6.40·26-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (−0.0984 + 0.170i)5-s + 0.353i·8-s − 0.139i·10-s + (0.135 − 0.0784i)11-s + (−1.53 − 0.888i)13-s + (−0.125 − 0.216i)16-s − 0.0793·17-s − 0.972i·19-s + (0.0492 + 0.0852i)20-s + (−0.0554 + 0.0960i)22-s + (−0.261 − 0.151i)23-s + (0.480 + 0.832i)25-s + 1.25·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.147 - 0.988i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.147 - 0.988i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7959629256\)
\(L(\frac12)\) \(\approx\) \(0.7959629256\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.220 - 0.381i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.450 + 0.260i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.55 + 3.20i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 0.327T + 17T^{2} \)
19 \( 1 + 4.23iT - 19T^{2} \)
23 \( 1 + (1.25 + 0.725i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.74 - 3.31i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-6.07 - 3.50i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 3.68T + 37T^{2} \)
41 \( 1 + (2.96 - 5.13i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.21 - 9.03i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.02 - 6.97i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 7.95iT - 53T^{2} \)
59 \( 1 + (-2.45 + 4.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.33 + 0.771i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.26 - 5.65i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 16.2iT - 71T^{2} \)
73 \( 1 - 4.12iT - 73T^{2} \)
79 \( 1 + (-0.662 - 1.14i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.55 + 14.8i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 11.7T + 89T^{2} \)
97 \( 1 + (10.6 - 6.12i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.073305455695238834155179522908, −8.263973166307524637795692330475, −7.48640157600280249229949539770, −7.02688377114342051659684316278, −6.11514914682360265389513712817, −5.19891211894828725450672317433, −4.56768924318596683687947461123, −3.15914550039202470078754924680, −2.42524624421761607873348039778, −0.991096002444467319554657937452, 0.37479309074308555550342390143, 1.88083493192270947513988543608, 2.56463661055296688960908660503, 3.87306120605792020552423948196, 4.50459170890432988840462080517, 5.58445916397406422508507613925, 6.50748934038056720275358056536, 7.36012405839420032043295835476, 7.86180818683014211798337300717, 8.800539966619166823178958706936

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