Properties

Label 2-2646-63.20-c1-0-15
Degree $2$
Conductor $2646$
Sign $0.137 - 0.990i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (1.77 + 3.07i)5-s − 0.999i·8-s − 3.55i·10-s + (2.61 + 1.51i)11-s + (0.888 − 0.513i)13-s + (−0.5 + 0.866i)16-s + 1.61·17-s + 8.22i·19-s + (−1.77 + 3.07i)20-s + (−1.51 − 2.61i)22-s + (2.90 − 1.67i)23-s + (−3.80 + 6.59i)25-s − 1.02·26-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.794 + 1.37i)5-s − 0.353i·8-s − 1.12i·10-s + (0.789 + 0.455i)11-s + (0.246 − 0.142i)13-s + (−0.125 + 0.216i)16-s + 0.392·17-s + 1.88i·19-s + (−0.397 + 0.687i)20-s + (−0.322 − 0.558i)22-s + (0.606 − 0.350i)23-s + (−0.761 + 1.31i)25-s − 0.201·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.137 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2646\)    =    \(2 \cdot 3^{3} \cdot 7^{2}\)
Sign: $0.137 - 0.990i$
Analytic conductor: \(21.1284\)
Root analytic conductor: \(4.59656\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2646} (1763, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2646,\ (\ :1/2),\ 0.137 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.613836910\)
\(L(\frac12)\) \(\approx\) \(1.613836910\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-1.77 - 3.07i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.61 - 1.51i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.888 + 0.513i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 1.61T + 17T^{2} \)
19 \( 1 - 8.22iT - 19T^{2} \)
23 \( 1 + (-2.90 + 1.67i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.70 - 2.13i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.18 - 2.99i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 5.84T + 37T^{2} \)
41 \( 1 + (-0.0472 - 0.0817i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.05 + 5.29i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.57 + 4.45i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 3.18iT - 53T^{2} \)
59 \( 1 + (-4.42 - 7.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.06 - 2.34i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.187 - 0.325i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.9iT - 71T^{2} \)
73 \( 1 - 1.31iT - 73T^{2} \)
79 \( 1 + (0.462 - 0.800i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.43 + 9.40i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.70T + 89T^{2} \)
97 \( 1 + (13.3 + 7.69i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.159934710743409988566383715043, −8.386585503601489954482751709652, −7.39815728120119069816421440320, −6.85609336680426290674912181692, −6.15629806669639813745928590520, −5.34032510227165000246338945185, −3.86380694347119323413265908490, −3.27160172050263890232037217550, −2.21492256544877322231278603778, −1.40881136799783741983543026209, 0.70148581911769465724449474168, 1.47890734304307581594993500109, 2.67377459391663514351321009996, 4.06179862782601753935216658106, 5.00230866857150821065634831803, 5.57002940280703881023634573670, 6.43387434070688313030129254184, 7.13117718680003572649275393667, 8.200173233417395952913241602848, 8.788072492196908017427494643930

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