Properties

Label 2-21-7.5-c4-0-2
Degree $2$
Conductor $21$
Sign $0.712 - 0.701i$
Analytic cond. $2.17076$
Root an. cond. $1.47335$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (4.5 − 2.59i)3-s + (6 + 10.3i)4-s + (9 + 5.19i)5-s + 10.3i·6-s + (38.5 + 30.3i)7-s − 56·8-s + (13.5 − 23.3i)9-s + (−18 + 10.3i)10-s + (−97 − 168. i)11-s + (54 + 31.1i)12-s − 164. i·13-s + (−91 + 36.3i)14-s + 54·15-s + (−40 + 69.2i)16-s + (−210 + 121. i)17-s + ⋯
L(s)  = 1  + (−0.250 + 0.433i)2-s + (0.5 − 0.288i)3-s + (0.375 + 0.649i)4-s + (0.359 + 0.207i)5-s + 0.288i·6-s + (0.785 + 0.618i)7-s − 0.875·8-s + (0.166 − 0.288i)9-s + (−0.179 + 0.103i)10-s + (−0.801 − 1.38i)11-s + (0.375 + 0.216i)12-s − 0.973i·13-s + (−0.464 + 0.185i)14-s + 0.239·15-s + (−0.156 + 0.270i)16-s + (−0.726 + 0.419i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.712 - 0.701i$
Analytic conductor: \(2.17076\)
Root analytic conductor: \(1.47335\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :2),\ 0.712 - 0.701i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.32564 + 0.542665i\)
\(L(\frac12)\) \(\approx\) \(1.32564 + 0.542665i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.5 + 2.59i)T \)
7 \( 1 + (-38.5 - 30.3i)T \)
good2 \( 1 + (1 - 1.73i)T + (-8 - 13.8i)T^{2} \)
5 \( 1 + (-9 - 5.19i)T + (312.5 + 541. i)T^{2} \)
11 \( 1 + (97 + 168. i)T + (-7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + 164. iT - 2.85e4T^{2} \)
17 \( 1 + (210 - 121. i)T + (4.17e4 - 7.23e4i)T^{2} \)
19 \( 1 + (-226.5 - 130. i)T + (6.51e4 + 1.12e5i)T^{2} \)
23 \( 1 + (-56 + 96.9i)T + (-1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 - 1.04e3T + 7.07e5T^{2} \)
31 \( 1 + (1.00e3 - 582. i)T + (4.61e5 - 7.99e5i)T^{2} \)
37 \( 1 + (-537.5 + 930. i)T + (-9.37e5 - 1.62e6i)T^{2} \)
41 \( 1 - 1.30e3iT - 2.82e6T^{2} \)
43 \( 1 + 1.08e3T + 3.41e6T^{2} \)
47 \( 1 + (-1.87e3 - 1.08e3i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + (-1.10e3 - 1.90e3i)T + (-3.94e6 + 6.83e6i)T^{2} \)
59 \( 1 + (4.63e3 - 2.67e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-606 - 349. i)T + (6.92e6 + 1.19e7i)T^{2} \)
67 \( 1 + (1.18e3 + 2.05e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 8.93e3T + 2.54e7T^{2} \)
73 \( 1 + (-7.90e3 + 4.56e3i)T + (1.41e7 - 2.45e7i)T^{2} \)
79 \( 1 + (4.07e3 - 7.05e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 - 6.67e3iT - 4.74e7T^{2} \)
89 \( 1 + (-1.18e4 - 6.82e3i)T + (3.13e7 + 5.43e7i)T^{2} \)
97 \( 1 + 3.49e3iT - 8.85e7T^{2} \)
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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

−17.78340213118514644992673709190, −16.20309404186310495517491317802, −15.16541431360247892732185662370, −13.75806710007529918113110688400, −12.38183352896072947858929958054, −10.86756305348377399011198434557, −8.687032698455008374623286530672, −7.83594857995684727959195736051, −5.93617298363017033538052244961, −2.81921384214715572436655270343, 1.97587835613929894864156584210, 4.86556202543996125709901766866, 7.23153127298501343977157287942, 9.240818625419382141274034358616, 10.32117975401905520903699440027, 11.62602351913222686438257381884, 13.50148474438913402447037482135, 14.71232519218962986372349419454, 15.77343176117851996270227029567, 17.46974214424617603279787188109

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