Properties

Label 2-2646-63.4-c1-0-3
Degree $2$
Conductor $2646$
Sign $-0.0781 - 0.996i$
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 + 2.03·5-s − 0.999·8-s + (1.01 − 1.75i)10-s − 4·11-s + (−2.12 + 3.67i)13-s + (−0.5 + 0.866i)16-s + (−0.707 + 1.22i)17-s + (−0.398 − 0.690i)19-s + (−1.01 − 1.75i)20-s + (−2 + 3.46i)22-s − 6.74·23-s − 0.872·25-s + (2.12 + 3.67i)26-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.908·5-s − 0.353·8-s + (0.321 − 0.556i)10-s − 1.20·11-s + (−0.588 + 1.01i)13-s + (−0.125 + 0.216i)16-s + (−0.171 + 0.297i)17-s + (−0.0914 − 0.158i)19-s + (−0.227 − 0.393i)20-s + (−0.426 + 0.738i)22-s − 1.40·23-s − 0.174·25-s + (0.416 + 0.720i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0781 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0781 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $-0.0781 - 0.996i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ -0.0781 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7787042769\)
\(L(\frac12)\) \(\approx\) \(0.7787042769\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.03T + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + (2.12 - 3.67i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (0.707 - 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.398 + 0.690i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.74T + 23T^{2} \)
29 \( 1 + (-4.43 - 7.68i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.73 + 4.74i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.87 - 8.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.82 - 4.89i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.43 + 7.68i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.44 - 5.96i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.30 - 9.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.32 + 2.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.398 + 0.690i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.436 + 0.756i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 2.12T + 71T^{2} \)
73 \( 1 + (-7.68 + 13.3i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.93 - 5.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.15 + 7.19i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.53 + 6.12i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.92 + 15.4i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.297453470752229920307778220243, −8.395667021565303184200825736008, −7.58502141015411846077352744907, −6.51464218686823965842812349727, −5.93665199509718677787520553848, −5.00534010969253679348692982959, −4.44159501740290682296157860121, −3.22574868168804037214332036483, −2.31428526509118981838689612162, −1.63331271094752114612845717653, 0.19815863951540491637027979659, 2.09118273767723836306348849653, 2.81535614753016922363766570043, 3.97101342782201460718616011050, 5.03614992021055862943023494913, 5.54680080744680374882899788942, 6.19236887152940926430431411862, 7.10302252041285114198834000126, 8.009285895039084831601900775833, 8.285105313873410247597283097384

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