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Properties

Label 2-2646-63.38-c1-0-25
Degree $2$
Conductor $2646$
Sign $-0.616 + 0.787i$
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 + (−1.17 + 2.03i)5-s + i·8-s + (2.03 + 1.17i)10-s + (−4.91 + 2.83i)11-s + (1.48 − 0.859i)13-s + 16-s + (−0.884 + 1.53i)17-s + (−0.986 + 0.569i)19-s + (1.17 − 2.03i)20-s + (2.83 + 4.91i)22-s + (−3.18 − 1.83i)23-s + (−0.259 − 0.449i)25-s + (−0.859 − 1.48i)26-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (−0.525 + 0.909i)5-s + 0.353i·8-s + (0.643 + 0.371i)10-s + (−1.48 + 0.855i)11-s + (0.413 − 0.238i)13-s + 0.250·16-s + (−0.214 + 0.371i)17-s + (−0.226 + 0.130i)19-s + (0.262 − 0.454i)20-s + (0.605 + 1.04i)22-s + (−0.663 − 0.383i)23-s + (−0.0519 − 0.0899i)25-s + (−0.168 − 0.292i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.616 + 0.787i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.616 + 0.787i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.616 + 0.787i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.616 + 0.787i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5545341649\)
\(L(\frac12)\) \(\approx\) \(0.5545341649\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.17 - 2.03i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.91 - 2.83i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.48 + 0.859i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.884 - 1.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.986 - 0.569i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.18 + 1.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.59 + 2.07i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 8.37iT - 31T^{2} \)
37 \( 1 + (-4.59 - 7.96i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.99 + 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.76 + 3.04i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 2.22T + 59T^{2} \)
61 \( 1 + 8.99iT - 61T^{2} \)
67 \( 1 - 10.8T + 67T^{2} \)
71 \( 1 + 4.52iT - 71T^{2} \)
73 \( 1 + (4.62 + 2.67i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + (-6.27 + 10.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.580 + 1.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.97 + 2.29i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.520546591232294018052732970435, −7.81387224169799348967472018261, −7.30096808188747937509535579545, −6.23547579326033466539445196671, −5.40541741627640994832867520760, −4.38982514060855976397505221273, −3.68168241352350659000037233738, −2.68278103494930653208700852396, −2.02849345889702180607265584688, −0.21583020218528000162532449398, 0.982437460363756513479100445768, 2.58985706828773136766246898854, 3.71866944654435918618058661479, 4.54797196020630685140922326463, 5.35429792268025285679772175997, 5.86979013000109854834732437456, 6.97356748334283190718544045284, 7.70631066563990327888209327073, 8.394670888260581436440097465166, 8.780750029009840271944733548463

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