Properties

Label 2-21-7.2-c29-0-38
Degree $2$
Conductor $21$
Sign $0.791 - 0.611i$
Analytic cond. $111.883$
Root an. cond. $10.5775$
Motivic weight $29$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.68e4 − 2.92e4i)2-s + (−2.39e6 − 4.14e6i)3-s + (−3.02e8 − 5.23e8i)4-s + (8.22e9 − 1.42e10i)5-s − 1.61e11·6-s + (−1.78e12 + 1.26e11i)7-s + (−2.26e12 − 0.00195i)8-s + (−1.14e13 + 1.98e13i)9-s + (−2.77e14 − 4.81e14i)10-s + (−5.67e14 − 9.82e14i)11-s + (−1.44e15 + 2.50e15i)12-s − 1.28e16·13-s + (−2.65e16 + 5.44e16i)14-s − 7.86e16·15-s + (1.23e17 − 2.14e17i)16-s + (−4.59e17 − 7.96e17i)17-s + ⋯
L(s)  = 1  + (0.728 − 1.26i)2-s + (−0.288 − 0.499i)3-s + (−0.562 − 0.974i)4-s + (0.602 − 1.04i)5-s − 0.841·6-s + (−0.997 + 0.0703i)7-s − 0.182·8-s + (−0.166 + 0.288i)9-s + (−0.878 − 1.52i)10-s + (−0.450 − 0.779i)11-s + (−0.324 + 0.562i)12-s − 0.905·13-s + (−0.638 + 1.31i)14-s − 0.695·15-s + (0.429 − 0.744i)16-s + (−0.662 − 1.14i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(30-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+29/2) \, L(s)\cr =\mathstrut & (0.791 - 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(21\)    =    \(3 \cdot 7\)
Sign: $0.791 - 0.611i$
Analytic conductor: \(111.883\)
Root analytic conductor: \(10.5775\)
Motivic weight: \(29\)
Rational: no
Arithmetic: yes
Character: $\chi_{21} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 21,\ (\ :29/2),\ 0.791 - 0.611i)\)

Particular Values

\(L(15)\) \(\approx\) \(1.141039482\)
\(L(\frac12)\) \(\approx\) \(1.141039482\)
\(L(\frac{31}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (2.39e6 + 4.14e6i)T \)
7 \( 1 + (1.78e12 - 1.26e11i)T \)
good2 \( 1 + (-1.68e4 + 2.92e4i)T + (-2.68e8 - 4.64e8i)T^{2} \)
5 \( 1 + (-8.22e9 + 1.42e10i)T + (-9.31e19 - 1.61e20i)T^{2} \)
11 \( 1 + (5.67e14 + 9.82e14i)T + (-7.93e29 + 1.37e30i)T^{2} \)
13 \( 1 + 1.28e16T + 2.01e32T^{2} \)
17 \( 1 + (4.59e17 + 7.96e17i)T + (-2.40e35 + 4.17e35i)T^{2} \)
19 \( 1 + (-2.26e18 + 3.92e18i)T + (-6.06e36 - 1.05e37i)T^{2} \)
23 \( 1 + (2.55e19 - 4.42e19i)T + (-1.54e39 - 2.67e39i)T^{2} \)
29 \( 1 + 4.23e20T + 2.56e42T^{2} \)
31 \( 1 + (-2.69e21 - 4.66e21i)T + (-8.88e42 + 1.53e43i)T^{2} \)
37 \( 1 + (-2.09e22 + 3.62e22i)T + (-1.50e45 - 2.60e45i)T^{2} \)
41 \( 1 + 4.74e22T + 5.89e46T^{2} \)
43 \( 1 + 7.57e23T + 2.34e47T^{2} \)
47 \( 1 + (-7.82e23 + 1.35e24i)T + (-1.54e48 - 2.68e48i)T^{2} \)
53 \( 1 + (-3.84e24 - 6.65e24i)T + (-5.04e49 + 8.74e49i)T^{2} \)
59 \( 1 + (1.15e25 + 1.99e25i)T + (-1.13e51 + 1.95e51i)T^{2} \)
61 \( 1 + (6.49e25 - 1.12e26i)T + (-2.97e51 - 5.15e51i)T^{2} \)
67 \( 1 + (-7.51e25 - 1.30e26i)T + (-4.52e52 + 7.82e52i)T^{2} \)
71 \( 1 - 2.10e26T + 4.85e53T^{2} \)
73 \( 1 + (-5.61e26 - 9.73e26i)T + (-5.43e53 + 9.41e53i)T^{2} \)
79 \( 1 + (4.21e26 - 7.30e26i)T + (-5.37e54 - 9.30e54i)T^{2} \)
83 \( 1 - 3.75e27T + 4.50e55T^{2} \)
89 \( 1 + (-9.21e27 + 1.59e28i)T + (-1.70e56 - 2.95e56i)T^{2} \)
97 \( 1 - 1.07e29T + 4.13e57T^{2} \)
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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

−11.45879706772853600707058015594, −10.03091877572254686036240759264, −9.038489880781004629356109640957, −7.15703439334809524436153104497, −5.53837432000792638360481319811, −4.79470696320658361078377290937, −3.14989108476554774577036919709, −2.29545820439428667926418862409, −1.01135020344133378457649935329, −0.20238909808409036238286190123, 2.19750684933485721337394063536, 3.57342637570723354751809203199, 4.76815379555626359146295807826, 6.07521317766265612746603632919, 6.58673644448065560146042546694, 7.82149468832192326399410247143, 9.817611521247380537722691293549, 10.45817954649281979911784279406, 12.41323810778526581730673286136, 13.55904421141238609418315203533

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