↑

Properties

Label 2-2646-63.16-c1-0-27
Degree $2$
Conductor $2646$
Sign $0.740 + 0.672i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−0.499 + 0.866i)4-s + 4.37·5-s + 0.999·8-s + (−2.18 − 3.78i)10-s + 1.37·11-s + (1 + 1.73i)13-s + (−0.5 − 0.866i)16-s + (−0.686 − 1.18i)17-s + (2.5 − 4.33i)19-s + (−2.18 + 3.78i)20-s + (−0.686 − 1.18i)22-s + 1.62·23-s + 14.1·25-s + (0.999 − 1.73i)26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + 1.95·5-s + 0.353·8-s + (−0.691 − 1.19i)10-s + 0.413·11-s + (0.277 + 0.480i)13-s + (−0.125 − 0.216i)16-s + (−0.166 − 0.288i)17-s + (0.573 − 0.993i)19-s + (−0.488 + 0.846i)20-s + (−0.146 − 0.253i)22-s + 0.339·23-s + 2.82·25-s + (0.196 − 0.339i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.740 + 0.672i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.740 + 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.348107421\)
\(L(\frac12)\) \(\approx\) \(2.348107421\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.37T + 5T^{2} \)
11 \( 1 - 1.37T + 11T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.686 + 1.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 + 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 1.62T + 23T^{2} \)
29 \( 1 + (4.37 - 7.57i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1 + 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.31 + 4.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.05 + 7.02i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.37 + 7.57i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.05 + 8.76i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.55 - 2.69i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.05 + 1.83i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.11T + 71T^{2} \)
73 \( 1 + (-6.05 - 10.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.55 - 4.43i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.74 - 15.1i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (7.37 - 12.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.05 - 7.02i)T + (-48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.153637747691186541042817302763, −8.362812135901304168924439841501, −6.90873856850160907813397199365, −6.73418631536846687927658359386, −5.45976726159781777274951267038, −5.08232092036376915705442079312, −3.77488010739205293521573703460, −2.70841245137889764704750185484, −1.96210827240248144233764480101, −1.06531861506805654938919121009, 1.15184697686183789781344415282, 1.98444712853950827102541954376, 3.10961087710691822388791736826, 4.43023652852728076728308959737, 5.43145577194273731687321447054, 5.96429988730634482122502213035, 6.41564611820017212705639640631, 7.40149046301955337136296593719, 8.243780674974642562150238611470, 9.099898044167665547660821281436

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