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Properties

Label 2-2646-63.25-c1-0-21
Degree $2$
Conductor $2646$
Sign $0.925 - 0.378i$
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  + 2-s + 4-s + (0.686 − 1.18i)5-s + 8-s + (0.686 − 1.18i)10-s + (2.18 + 3.78i)11-s + (1 + 1.73i)13-s + 16-s + (2.18 − 3.78i)17-s + (2.5 + 4.33i)19-s + (0.686 − 1.18i)20-s + (2.18 + 3.78i)22-s + (−3.68 + 6.38i)23-s + (1.55 + 2.69i)25-s + (1 + 1.73i)26-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + (0.306 − 0.531i)5-s + 0.353·8-s + (0.216 − 0.375i)10-s + (0.659 + 1.14i)11-s + (0.277 + 0.480i)13-s + 0.250·16-s + (0.530 − 0.918i)17-s + (0.573 + 0.993i)19-s + (0.153 − 0.265i)20-s + (0.466 + 0.807i)22-s + (−0.768 + 1.33i)23-s + (0.311 + 0.539i)25-s + (0.196 + 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.925 - 0.378i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.925 - 0.378i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (2125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.925 - 0.378i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.293235216\)
\(L(\frac12)\) \(\approx\) \(3.293235216\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.686 + 1.18i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.18 + 3.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.68 - 6.38i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.37 + 2.37i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.18 + 8.98i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.55 - 7.89i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-1.37 + 2.37i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 7.11T + 59T^{2} \)
61 \( 1 - 14.1T + 61T^{2} \)
67 \( 1 - 15.1T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 + (2.55 - 4.43i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + (-2.74 + 4.75i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.62 + 2.81i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.55 + 7.89i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.038915056423630982635915759042, −7.996242305647915461882642163731, −7.25037403445911115118950886328, −6.61268713170299342355745241354, −5.52699921765123812190623866130, −5.15280828906485086372156006203, −4.07718685687282128558697300582, −3.47837402260748621256511321829, −2.09703543465424747840778170307, −1.32952869191418380386762259140, 0.941251260912025324134330762483, 2.30876981189369369969499923336, 3.22661551106782145607400256439, 3.85582515365357181673587323575, 4.96592497167679995812423978278, 5.76071916716643508696130002262, 6.48052555864268543744039672561, 6.92588542464054611060612918103, 8.237884918824355736899354554226, 8.523104818717090867839962595480

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