Properties

Label 2-2646-63.58-c1-0-39
Degree $2$
Conductor $2646$
Sign $-0.888 - 0.458i$
Analytic cond. $21.1284$
Root an. cond. $4.59656$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + (−1.5 − 2.59i)5-s + 8-s + (−1.5 − 2.59i)10-s + (−1.5 + 2.59i)11-s + (−0.5 + 0.866i)13-s + 16-s + (−1.5 − 2.59i)17-s + (−3.5 + 6.06i)19-s + (−1.5 − 2.59i)20-s + (−1.5 + 2.59i)22-s + (−4.5 − 7.79i)23-s + (−2 + 3.46i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (−0.670 − 1.16i)5-s + 0.353·8-s + (−0.474 − 0.821i)10-s + (−0.452 + 0.783i)11-s + (−0.138 + 0.240i)13-s + 0.250·16-s + (−0.363 − 0.630i)17-s + (−0.802 + 1.39i)19-s + (−0.335 − 0.580i)20-s + (−0.319 + 0.553i)22-s + (−0.938 − 1.62i)23-s + (−0.400 + 0.692i)25-s + (−0.0980 + 0.169i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.888 - 0.458i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.888 - 0.458i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (1.5 + 2.59i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.5 - 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.5 + 7.79i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 8T + 31T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.5 - 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-1.5 - 2.59i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-5.5 - 9.52i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (1.5 - 2.59i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.350799668777998441207730296158, −7.68732714419464148488953201406, −6.87852161832390512043012319430, −5.95738229707246535965970626520, −5.09223055653503548243724302807, −4.39746723970510222075624008082, −3.94494665523605340093394526127, −2.58578685073989810568903193098, −1.59850526679908483794222324322, 0, 1.99442414545846189084049874956, 2.98134475442050176936067560392, 3.60481960106364340740353551097, 4.42394985463542690387726380921, 5.52500386476360807129813953531, 6.16543752610544978727109387986, 7.03143017555354386618551610216, 7.55713329745651211248237912120, 8.350162035565962120298005670892

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